Mathematics: Calculus Basics

Calculus
Calculus, also called analysis, is the branch of mathematics that studies the length, area, and volume of objects, and the rate of change of quantities (which can be expressed as slopes of curves). The two principal branches of calculus are differential and integral.

Differential calculus is based on derivatives and takes the form,


Integral calculus is based on integrals and takes the form,


Some of the basic ideas of calculus were utilized as far back in history as Archimedes. However, its modern forms were developed by Newton and Leibniz.

Limits
The limit of a function is represented by the notation
. It is read as 'the limit of f of x as x approaches a.' In many cases,
 will simply be equal to
, but not always.

Limits are important because some functions are not defined or are not easy to evaluate at certain values of x.

The limit at the point is said to exist only if the limit is the same when approached from the right side as from the left:
).

Notice the symbol by the a in each case.

When x approaches a from the right, it approaches from the positive end of the number line. When x approaches a from the left, it approaches from the negative end of the number line.

If the limit as x approaches a differs depending on the direction from which it approaches, then the limit does not exist at a. In other words, if
 does not equal
, then the limit does not exist at a.

The limit also does not exist if either of the one-sided limits does not exist.

Situations in which the limit does not exist include a function that jumps from one value to another at a, one that oscillates between two different values as x approaches a, or one that increases or decreases without bounds as x approaches a. If the limit you calculate has a value of
is any constant, this means the function goes to infinity and the limit does not exist. <br> <br> It is possible for two functions that do not have limits to be multiplied to get a new function that does have a limit. Just because two functions do not have limits, do not assume that the product will not have a limit.<br> <br> Direct Substitution<br> The first thing to try when looking for a limit is direct substitution. To find the limit of a function <br><img data-cke-saved-src=" /> by direct substitution, substitute the value of a for x in the function and solve.

The following patterns apply to finding the limit of a function by direct substitution:












You can also use substitution for finding the limit of a trigonometric function, a polynomial function, or a rational function. Be sure that in manipulating an expression to find a limit that you do not divide by terms equal to zero.

In finding the limit of a composite function, begin by finding the limit of the innermost function. For example, to find
, first find the value of
. Then substitute this value for x in
 and solve.
The result is the limit of the original problem.

Limits and Operations
When finding the limit of the sum or difference of two functions, find the limit of each individual function and then add or subtract the results.

For example,
.
To find the limit of the product or quotient of two functions, find the limit of each individual function and then multiply or divide the results.

For example,
 and
, where
 and
. When finding the quotient of the limits of two functions, make sure the denominator is not equal to zero. If it is, use differentiation or L'Hôpital's rule to find the limit.

To find the limit of a power of a function or a root of a function, find the limit of the function and then raise the limit to the original power or take the root of the limit.

For example,
is a positive integer and <br><img data-cke-saved-src=" /> for all even values of n.

To find the limit of a function multiplied by a scalar, find the limit of the function and multiply the result by the scalar. For example,
is a real number.<br> <br> L" /> Sometimes solving
 by the direct substitution method will result in the numerator and denominator both being equal to zero, or both being equal to infinity. This outcome is called an indeterminate form. The limit cannot be directly found by substitution in these cases. L'Hôpital's rule is a useful method for finding the limit of a problem in the indeterminate form. L'Hôpital's rule allows you to find the limit using derivatives. Assuming both the numerator and denominator are differentiable, and that both are equal to zero when the direct substitution method is used, take the derivative of both the numerator and the denominator and then use the direct substitution method.


For example, if
, take the derivatives of
 and
and then find
. If
, then you have found the limit of the original function. If
 and
, L'Hôpital's rule may be applied to the function
, and so on until either a limit is found, or it can be determined that the limit does not exist.

Squeeze Theorem
The squeeze theorem is known by many names, including the sandwich theorem, the sandwich rule, the squeeze lemma, the squeezing theorem, and the pinching theorem. No matter what you call it, the principle is the same. To prove the limit of a difficult function exists, find the limits of two functions, one on either side of the unknown, that are easy to compute. If the limits of these functions are equal, then that is also the limit of the unknown function. In mathematical terms, the theorem is:
 for all values of x where
 is the function with the unknown limit, and if
, then this limit is also equal to


To find the limit of an expression containing an absolute value sign, take the absolute value of the limit. If
is the numerical value for the limit, then <br><img data-cke-saved-src=" />. Also, if
, then
. The trick comes when you are asked to find the limit as n approaches from the left. Whenever the limit is being approached from the left, it is being approached from the negative end of the domain. The absolute value sign makes everything in the equation positive, essentially eliminating the negative side of the domain. In this case, rewrite the equation without the absolute value signs and add a negative sign in front of the expression. For example,
 becomes
.

Derivatives
The derivative of a function is a measure of how much that function is changing at a specific point, and is the slope of a line tangent to a curve at the specific point. The derivative of a function

,
,
,
 , and
. The definition of the derivative of a function is
. However, this formula is rarely used.

There is a simpler method you can use to find the derivative of a polynomial.

Given a function
, multiply each exponent by its corresponding coefficient to get the new coefficient and reduce the value of the exponent by one. Coefficients with no variable are dropped. This gives
, a pattern that can be repeated for each successive derivative.

Differentiable functions are functions that have a derivative. Some basic rules for finding derivatives of functions are:















This last formula is also known as the chain rule. If you are finding the derivative of a polynomial that is raised to a power, let the polynomial be represented by
 and use the chain rule. The chain rule is one of the most important concepts to grasp in the early stages of learning calculus. Many other rules and shortcuts are based upon the chain rule.

Difference Quotient and Derivative
A secant is a line that connects two points on a curve. The difference quotient gives the slope of an arbitrary secant line that connects the point
 with a nearby point
 on the graph of the function f. The difference quotient is the same formula that is always used to determine a slope—the change in y divided by the change in x. It is written as
.

A tangent is a line that touches a curve at one point. The tangent and the curve have the same slope at the point where they touch. The derivative is the function that gives the slope of both the tangent and the curve of the function at that point. The derivative is written as the limit of the difference quotient, or:

If the function is f, the derivative is denoted as
, and it is the slope of the function f at point
. It is expressed as:



Implicit Functions
An implicit function is one where it is impossible, or very difficult, to express one variable in terms of another by normal algebraic methods. This would include functions that have both variables raised to a power greater than 1, functions that have two variables multiplied by each other, or a combination of the two. To differentiate such a function with respect to x, take the derivative of each term that contains a variable, either x or y. When differentiating a term with y, use the chain rule, first taking the derivative with respect to y, and then multiplying by
. If a term contains both x and y, you will have to use the product rule as well as the chain rule. Once the derivative of each individual term has been found, use the rules of algebra to solve for
 to get the final answer.

Derivatives of Trigonometric Functions
Trigonometric functions are any functions that include one of the six trigonometric expressions. The following rules for derivatives apply for all trigonometric differentiation:

For functions that are a combination of trigonometric and algebraic expressions, use the chain rule:












Functions involving the inverses of the trigonometric functions can also be differentiated.










In each of the above expressions, u represents a differentiable function. Also, the value of u must be such that the radicand, if applicable, is a positive number. Remember the expression
 means to take the derivative of the function u with respect to the variable x.

Derivatives of Exponential and Logarithmic Functions
Exponential functions are in the form
, which has itself as its derivative:
. For functions that have a function as the exponent rather than just an x, use the formula
. The inverse of the exponential function is the natural logarithm. To find the derivative of the natural logarithm, use the formula
 .

If you are trying to solve an expression with a variable in the exponent, use the formula
is a positive real number and <i>x</i> is any real number. To find the derivative of a function in this format, use the formula <br><img data-cke-saved-src=" />. If the exponent is a function rather than a single variable x, use the formula
. If you are trying to solve an expression involving a logarithm, use the formula
 or
.

Continuity
A function can be either continuous or discontinuous. A conceptual way to describe continuity is this: A function is continuous if its graph can be traced with a pen without lifting the pen from the page. In other words, there are no breaks or gaps in the graph of the function. However, this is only a description, not a technical definition. A function is continuous at the point
 if the three following conditions are met:

1.
 is defined
2.
 exists
3.

If any of these conditions are not met, the function is discontinuous at the point
.
A function can be continuous at a point, continuous over an interval, or continuous everywhere. The above rules define continuity at a point. A function that is continuous over an interval
 is continuous at the points a and b and at every point between them. A function that is continuous everywhere is continuous for every real number, that is, for all points in its domain.

Discontinuity
Discontinuous functions are categorized according to the type or cause of discontinuity. Three examples are point, infinite, and jump discontinuity. A function with a point discontinuity has one value of x for which it is not continuous. A function with infinite discontinuity has a vertical asymptote at
 and
 is undefined. It is said to have an infinite discontinuity at
.
A function with jump discontinuity has one-sided limits from the left and from the right, but they are not equal to one another, that is,
. It is said to have a jump discontinuity at
.

The function,
, plotted in the graph has an infinite discontinuity. It has a vertical asymptote at
, as such, the function is undefined at
.

Differentiability
A function is said to be differentiable at point
if it has a derivative at that point, that is, if  exists. For a function to be differentiable, it must be continuous because the slope cannot be defined at a point of discontinuity. Furthermore, for a function to be differentiable, its graph must not have any sharp turn for which it is impossible to draw a tangent line. The sine function is an example of a differentiable function. It is continuous, and a tangent line can be drawn anywhere along its graph.

The absolute value function,
, is an example of a function that is not differentiable:

It is continuous, but it has a sharp turn at  which prohibits the drawing of a tangent at that point. All differentiable functions are continuous, but not all continuous functions are differentiable, as the absolute value function demonstrates.
The function
is not differentiable because it is not continuous. It has a discontinuity at
.
Therefore, a tangent could not be drawn at that point.

Approximating a Derivative from a Table of Values
The derivative of a function at a particular point is equal to the slope of the graph of the function at that point. For a nonlinear function, it can be thought of as the limit of the slope of a line drawn between two other points on the function as those points become closer to the point in question. Such a line drawn through two points on the function is called a secant of the function.
This definition of the derivative in terms of the secant allows us to approximate the derivative of a function at a point from a table of values: we take the slope of the line through the points on either side.

That is, if the point lies between
 and
, the slope of the secant—the approximate derivative—is
. (This is also equal to the average slope over the interval
.)

For example, consider the function represented by the following table:






Suppose we want to know the derivative of the function when
.
This lies between the points
 and
; the approximate derivative is
.

Position, Velocity, and Acceleration
Velocity is a specific type of rate of change. It refers to the rate of change of the position of an object with relation to a reference frame. Acceleration is the rate of change of velocity.

Average velocity over a period of time is found using the formula
 , where
 and
 are specific points in time and
 and
 are the distances traveled at those points in time.

Instantaneous velocity at a specific time, t, is found using the limit
, or
.

Remember that velocity at a given point is found using the first derivative, and acceleration at a given point is found using the second derivative. Therefore, the formula for acceleration at a given point in time is found using the formula
is velocity, and<i> s</i> is distance or location.<br> <br> Using First and Second Derivatives<br> The first derivative of a function is equal to the rate of change of the function. The sign of the rate of change shows whether the value of the function is increasing or decreasing. A positive rate of change—and therefore a positive first derivative—represents that the function is increasing at that point. A negative rate of change represents that the function is decreasing. If the rate of change is zero, the function is not changing, i.e. it is constant.<br> <br> For example, consider the function <br><img data-cke-saved-src=" />. The derivative of this function is
. This derivative is a quadratic function with zeroes at
 and
; by plugging in points in each interval we can find that
 is positive when
 and when
 and negative when
. Thus
 is increasing in the interval
 and decreasing in the interval
.

Extrema
The maximum and minimum values of a function are collectively called the extrema of the function. Both maxima and minima can be local, also known as relative, or absolute. A local maximum or minimum refers to the value of a function near a certain value of x. An absolute maximum or minimum refers to the value of a function on a given interval.

The local maximum of a function is the largest value that the function attains near a certain value of x. For example, function f has a local maximum at
 if
 is the largest value that f attains as it approaches b.

Conversely, the local minimum is the smallest value that the function attains near a certain value of x. In other words, function f has a local minimum at
 if
 is the smallest value that f attains as it approaches b.

The absolute maximum of a function is the largest value of the function over a certain interval. The function f has an absolute maximum at
 if
 for all x in the domain of f.

The absolute minimum of a function is the smallest value of the function over a certain interval. The function f has an absolute minimum at
 if
 for

Critical Points
Remember Rolle's theorem, which states that if two points have the same value in the range that there must be a point between them where the slope of the graph is zero. This point is located at a peak or valley on the graph. A peak is a maximum point, and a valley is a minimum point. The relative minimum is the lowest point on a graph for a given section of the graph. It may or may not be the same as the absolute minimum, which is the lowest point on the entire graph. The relative maximum is the highest point on one section of the graph. Again, it may or may not be the same as the absolute maximum. A relative extremum (plural extrema) is a relative minimum or relative maximum point on a graph.

A critical point is a point
 that is part of the domain of a function, such that either
 or
 does not exist. If either of these conditions is true, then x is either an inflection point or a point at which the slope of the curve changes sign. If the slope changes sign, then a relative minimum or maximum occurs.

In graphing an equation with relative extrema, use a sign diagram to approximate the shape of the graph. Once you have determined the relative extrema, calculate the sign of a point on either side of each critical point. This will give a general shape of the graph, and you will know whether each critical point is a relative minimum, a relative maximum, or a point of inflection.

First Derivative Test
Remember that critical points occur where the slope of the curve is 0. Also remember that the first derivative of a function gives the slope of the curve at a particular point on the curve. Because of this property of the first derivative, the first derivative test can be used to determine if a critical point is a minimum or maximum. If
 is negative at a point to the left of a critical number and
 is positive at a point to the right of a critical number, then the critical number is a relative minimum. If
 is positive to the left of a critical number and
 is negative to the right of a critical number, then the critical number is a relative maximum. If
 has the same sign on both sides, then the critical number is a point of inflection.

Second Derivative Test
The second derivative, designated by , is helpful in determining whether the relative extrema of a function are relative maximums or relative minimums. If the second derivative at the critical point is greater than zero, the critical point is a relative minimum. If the second derivative at the critical point is less than zero, the critical point is a relative maximum. If the second derivative at the critical point is equal to zero, you must use the first derivative test to determine whether the point is a relative minimum or a relative maximum.

There are a couple of ways to determine the concavity of the graph of a function. To test a portion of the graph that contains a point with domain p, find the second derivative of the function and evaluate it for p. If
, then the graph is concave upward at that point. If
, then the graph is concave downward at that point.

The point of inflection on the graph of a function is the point at which the concavity changes from concave downward to concave upward or from concave upward to concave downward.

The easiest way to find the points of inflection is to find the second derivative of the function and then solve the equation
. Remember that if
, the graph is concave upward, and if
, the graph is concave downward. Logically, the concavity changes at the point when
:


The derivative tests that have been discussed thus far can help you get a rough picture of what the graph of an unfamiliar function looks like. Begin by solving the equation
 to find all the zeros of the function, if they exist. Plot these points on the graph. Then, find the first derivative of the function and solve the equation
 to find the critical points. Remember the numbers obtained here are the x portions of the coordinates. Substitute these values for x in the original function and solve for y to get the full coordinates of the points. Plot these points on the graph. Take the second derivative of the function and solve the equation
 to find the points of inflection. Substitute in the original function to get the coordinates and graph these points. Test points on both sides of the critical points to test for concavity and draw the curve.

Derivative Problems
A derivative represents the rate of change of a function; thus, derivatives are a useful tool for solving any problem that involves finding the rate at which a function is changing. In its simplest form, such a problem might provide a formula for a quantity as a function of time and ask for its rate of change at a particular time.

If the temperature in a chamber in degrees Celsius is equal to
is the time in seconds, then the derivative of the function represents the rate of change of the temperature over time. The rate of change is equal to <br><img data-cke-saved-src=" />, and the initial rate of change is
.

Suppose we are told that the net profit that a small company makes when it produces and sells x units of a product is equal to
. The derivative of this function would be the additional profit for each additional unit sold, a quantity known as the marginal profit. The marginal profit in this case is
.

Solving Related Rates Problems
A related rate problem is one in which one variable has a relation with another variable, and the rate of change of one of the variables is known. With that information, the rate of change of the other variable can be determined. The first step in solving related rates problems is defining the known rate of change. Then, determine the relationship between the two variables, then the derivatives (the rates of change), and finally substitute the problem's specific values.

Consider the following example:

The side of a cube is increasing at a rate of 2 feet per second. Determine the rate at which the volume of the cube is increasing when the side of the cube is 4 feet long.

For the problem in question, the known rate of change can be expressed as
is the length of the side and <i>t</i> is the elapsed time in seconds. The relationship between the two variables of the cube is <br><img src=" /> becomes


Now, the chain rule is applied to differentiate both sides of the equation with respect to t.


Finally, the specific value of
 feet is substituted, and the equation is evaluated.







Therefore, when a side of the cube is 4 feet long, the volume of the cube is increasing at a rate of
.

Solving Optimization Problems

An optimization problem is a problem in which we are asked to find the value of a variable that maximizes or minimizes a particular value. Because the maximum or maximum occurs at a critical point, and because the critical point occurs when the derivative of the function is zero, we can solve an optimization problem by setting the derivative of the function to zero and solving for the desired variable.

For example, suppose a farmer has 720 m of fencing, and wants to use it to fence in a 2 by 3 block of identical rectangular pens. What dimensions of the pens will maximize their area?

We can draw a diagram:


We want to maximize the area of each pen,
. However, we have the additional constraint that the farmer has only 720 m of fencing. In terms of x and y, we can count the number of segments of each length, 9 for x and 8 for y, so the total amount of fencing required will be
. Our constraint becomes
; solving for y yields
. We can substitute that into the area equation to get
. Taking the derivative yields
; setting that equal to zero and solving for x yields
, thus, the maximum dimensions of the pen are 40 by 45 meters.

Characteristics of Functions (Using Calculus)
Rolle's theorem states that if a differentiable function has two different values in the domain that correspond to a single value in the range, then the function must have a point between them where the slope of the tangent to the graph is zero. This point will be a maximum or a minimum value of the function between those two points.

The maximum or minimum point is the point at which
is within the appropriate interval of the function" />

Mean Value Theorem
According to the mean value theorem, between any two points on a curve, there exists a tangent to the curve whose slope is parallel to the chord formed by joining those two points.

Remember the formula for slope:
. In a function,  represents the value for y. Therefore, if you have two points on a curve, m and n, the corresponding points are
 and
.

Assuming
, the formula for the slope of the chord joining those two points is
 .

This must also be the slope of a line parallel to the chord, since parallel lines have equal slopes. Therefore, there must be a value p between m and n
such that
 .

For a function to have continuity, its graph must be an unbroken curve. That is, it is a function that can be graphed without having to lift the pencil to move it to a different point.

To say a function is continuous at point p, you must show the function satisfies three requirements. First,
 must exist.

If you evaluate the function at p, it must yield a real number.

Second, there must exist a relationship such that
. Finally, the following relationship must be true:


If all three of these requirements are met, a function is considered continuous at p. If any one of them is not true, the function is not continuous at p.

Tangents
Tangents are lines that touch a curve in exactly one point and have the same slope as the curve at that point. To find the slope of a curve at a given point and the slope of its tangent line at that point, find the derivative of the function of the curve.
If the slope is undefined, the tangent is a vertical line. If the slope is zero, the tangent is a horizontal line.

A line that is normal to a curve at a given point is perpendicular to the tangent at that point. Assuming , the equation for the normal line at point (a, b) is:
. The easiest way to find the slope of the normal is to take the negative reciprocal of the slope of the tangent. If the slope of the tangent is zero, the slope of the normal is undefined. If the slope of the tangent is undefined, the slope of the normal is zero.

Antiderivatives (Integrals)
The antiderivative of a function is the function whose first derivative is the original function. Antiderivatives are typically represented by capital letters, while their first derivatives are represented by lower case letters.

For example, if
is the antiderivative of <i>f.</i> Antiderivatives are also known as indefinite integrals. When taking the derivative of a function, any constant terms in the function are eliminated because their derivative is 0. To account for this possibility, when you take the indefinite integral of a function, you must add an unknown constant <i>C</i> to the end of the function. Because there is no way to know what the value of the original constant was when looking just at the first derivative, the integral is indefinite.<br> <br> To find the indefinite integral, reverse the process of differentiation. Below are the formulas for constants and powers of <i>x</i>.<br><img data-cke-saved-src=" />





Recall that in the differentiation of powers of x, you multiplied the coefficient of the term by the exponent of the variable and then reduced the exponent by one. In integration, the process is reversed: add one to the value of the exponent, and then divide the coefficient of the term by this number to get the integral.

Because you do not know the value of any constant term that might have been in the original function, add C to the end of the function once you have completed this process for each term.

Finding the integral of a function is the opposite of finding the derivative of the function. Where possible, you can use the trigonometric or logarithmic differentiation formulas in reverse, and add C to the end to compensate for the unknown term.
In instances where a negative sign appears in the differentiation formula, move the negative sign to the opposite side (multiply both sides by
) to reverse for the integration formula. You should end up with the following formulas:

Integration by substitution is the integration version of the chain rule for differentiation.

The formula for integration by substitution is given by the equation


When a function is in a format that is difficult or impossible to integrate using traditional integration methods and formulas due to multiple functions being combined, use the formula shown above to convert the function to a simpler format that can be integrated directly.

Integration by parts is the integration version of the product rule for differentiation. Whenever you are asked to find the integral of the product of two different functions or parts, integration by parts can make the process simpler.

Recall for differentiation
. This can also be written
, where
 and
. Rearranging to integral form gives the formula:




When using integration by parts, the key is selecting the best functions to substitute for u and v so that you make the integral easier to solve and not harder.

While the indefinite integral has an undefined constant added at the end, the definite integral can be calculated as an exact real number. To find the definite integral of a function over a closed interval, use the formula
is the integral of <i>f</i>. Because you have been given the boundaries of <i>n</i> and <i>m</i>, no undefined constant <i>C</i> is needed.<br> <br> First Fundamental Theorem of Calculus<br> The first fundamental theorem of calculus shows that the process of indefinite integration can be reversed by finding the first derivative of the resulting function. It also gives the relationship between differentiation and integration over a closed interval of the function. For example, assuming a function is continuous over the interval <br><img data-cke-saved-src=" />, you can find the definite integral by using the formula


To find the average value of the function over the given interval, use the formula:


Second Fundamental Theorem of Calculus
The second fundamental theorem of calculus is related to the first. This theorem states that, assuming the function is continuous over the interval you are considering, taking the derivative of the integral of a function will yield the original function.

The general format for this theorem for any point having a domain value equal to c in the given interval is:


For each of the following properties of integrals of function f, the variables m, n, and p represent values in the domain of the given interval of
.

The function is assumed to be integrable across all relevant intervals.
Swapping the limits of integration:



Function multiplied by a constant:

Separating the integral into parts:


If the limits of integration are equivalent:

If
 is an even function and the limits of integration are symmetric:

If
 is an odd


Matching Functions to Derivatives or Accumulations

Derivatives
We can use what we know about the meaning of a derivative to match the graph of a function with a graph of its derivative. For one thing, we know that where the function has a critical point, the derivative is zero.

Therefore, at every x value at which the graph of a function has a maximum or minimum, the derivative must cross the x axis—and conversely, everywhere the graph of the derivative crosses the x axis, the function must have a critical point: either a maximum, a minimum, or an inflection point. If this is still not enough to identify the correct match, we can also use the fact that the sign of the derivative corresponds to whether the function is increasing or decreasing: everywhere the graph of the derivative is above the x axis, the function must be increasing (its slope is positive), and everywhere the graph of the derivative is below the x axis, the function must be decreasing (its slope is negative).

For example, below are graphs of the function and its derivative. The maxima and minima of the function
(left) are circled, and the zeroes of the derivative (right) are circled.


Accumulations
The accumulation of a function is another name for its antiderivative, or integral. We can use the relationship between a function and its antiderivative to match the corresponding graphs. For example, we know that where the graph of the function is above the x axis, the function is positive, thus the accumulation must be increasing (its slope is positive); where the graph of the function is below the x axis, the accumulation must be decreasing (its slope is negative). It follows that where the function changes from positive to negative—where the graph crosses the x axis with a negative slope—, its accumulation changes from increasing to decreasing—so the accumulation has a local maximum. Where the function changes from negative to positive—where its graph crosses the x axis with a positive slope—, the accumulation has a local minimum. of a function and its accumulation. The points on the function (left) where the graph crosses the x axis are circled; the local minima and maxima of the accumulation (right) are circled.


Riemann Sums
A Riemann sum is a sum used to approximate the definite integral of a function over a particular interval by dividing the area under the function into vertical rectangular strips and adding the areas of the strips. The height of each strip is equal to the value of the function at some point within the interval covered by the strip. Formally, if we divide the interval over which we are finding the area into n intervals bounded by the n +1 points
 (where
 and
 are the left and right bounds of the interval), then the Riemann sum is
, where
 and
 is some point in the interval . In principle, any point in the interval can be chosen, but common choices include the left endpoint of the interval (yielding the left Riemann sum), the right endpoint (yielding the right Riemann sum), and the midpoint of the interval (the basis of the midpoint rule). Usually it is convenient to set all the intervals to the same width, although the definition of the Riemann sum does not require this.

The following graphic shows the rectangular strips used for one possible Riemann sum of a particular


Left and Right Riemann Sums
A Riemann sum is an approximation to the definite integral of a function over a particular interval performed by dividing it into smaller intervals and summing the products of the width of each interval and the value of the function evaluated at some point within the interval. The left

Riemann sum is a Riemann sum in which the function is evaluated at the left endpoint of each interval. In the right Riemann sum, the function is evaluated at the right endpoint of each interval.

When the function is increasing, the left Riemann sum will always underestimate the function. This is because we are evaluating the function at the minimum point within each interval; the integral of the function in the interval will be larger than the estimate. Conversely, the right Riemann sum is evaluating the function at the maximum point within each interval, thus it will always overestimate the function. Consider the following diagrams, in which the area under the same increasing function is shown approximated by a left Riemann sum and a right

Riemann sum:

For a decreasing function these considerations are reversed: a left Riemann sum will overestimate the integral, and a right Riemann sum will underestimate it.

Midpoint Rule
The midpoint rule is a way of approximating the definite integral of a function over an interval by dividing the interval into smaller sub-intervals, multiplying the width of each sub-interval by the value of the function at the midpoint of the sub-interval, and then summing these products. This is a special case of the Riemann sum, specifying the midpoint of the interval as the point at which the function is to be evaluated. The approximation found using the midpoint rule is usually more accurate than that found using the left or right Riemann sum, though as the number of intervals becomes very large the difference becomes negligible.

For example, suppose we are asked to estimate by the midpoint rule the integral of
 in the interval [2, 4]. We can divide this interval into four intervals of width
: [2, 2.5], [2.5, 3], [3, 3.5], and [3.5, 4]. (The more intervals, the more accurate the estimate, but we'll use a small number of intervals in this example to keep it simple.) The midpoint rule then gives an estimate of
, not far from the actual value of
.

Trapezoid Rule
The trapezoid rule is a method of approximating the definite integral of a function by dividing the area under the function into a series of trapezoidal strips, the upper corners of the trapezoid touching the function, and adding the areas of the strips.
The following diagram shows the use of the trapezoid rule to estimate the integral of the function
 in the interval
:


Mathematically, if we define the endpoints of the n subdivisions to be
, where
 and
 are the endpoints of the entire interval over which we are estimating the integral, then the result of the application of the trapezoid rule is equal to
. For the example shown above, that yields
, or 10.5—not far from the actual value of
. (Of course, we could have achieved more accuracy by using smaller subdivisions.)

The trapezoid rule is related to the Riemann sum, but usually gives more accurate results than the left or right Riemann sum for the same number of intervals. In fact, it isn't hard to prove that the answer given by the trapezoid rule is equal to the average of the left and right Riemann sums using the same partition.

Limit of Riemann Sums
As the number of sub-intervals becomes larger, and the width of each sub-interval becomes smaller, the approximation becomes increasingly accurate, and at the limit as the number of sub-intervals approaches infinity and their width approaches zero, the value becomes exact. In fact, the definite integral is often defined as a limit of Riemann sums.

It's possible to find the definite integral by this method.

Suppose we want to find the integral of
 over the interval [0, 2]. We'll divide this interval into n sub-intervals of equal width and evaluate the function at the right endpoint of each sub-interval.

(This choice is arbitrary; at the limit the answer would be the same if we chose the left endpoint, or any other point within the interval.) Our Riemann sum becomes
, thus this becomes
. At the limit as
, this becomes
. This is the same result as we get by integrating directly:
.

Uses for Integration

Calculating Distances
When given the velocity of an object over time, it's possible to find a distance by integration. The velocity is the rate of change of the position; therefore, the displacement is the accumulation of the velocity: that is, the integral of the velocity is the displacement. However, if asked to find the total distance traveled (as opposed to the displacement), it's important to take the sign into account: we must integrate not just the velocity, but the absolute value of the velocity, which essentially means integrating separately over each interval in which the velocity has a different sign.

For example, suppose we're asked to find the total distance traveled from
 to
 by an object moving with a velocity in meters per second given by the equation
. This function is zero when
 or 4.
 is positive when
 and negative when
. Thus, the distance travelled is
















Calculating Areas
One way to calculate the area of an irregular shape is to find a formula for the width of the shape along the x direction as a function of the y coordinate, and then integrate over y, or vice versa. What this amounts to is dividing the area into thin strips and adding the areas of the strips—and then taking the limit as the width of the strips approaches zero.

For example, suppose we want to find the area enclosed by the functions
 and
). The height of this enclosure is equal to
; we can find the area by integrating this height over x. The two shapes intersect at the points
 and
, thus our limits of integration are –1 and 1. Thus the area can be found as:


Calculating Volumes
One way to calculate the volume of a three-dimensional shape is to find a formula for its cross-sectional area perpendicular to some axis and then integrate over that axis. Effectively, this divides the shape into thin, flat slices and adds the volumes of the slices—and then takes the limit as the thickness of the slices approaches zero.

For example, suppose we want to find the area of the ellipsoid
. If we take a cross-section parallel to the z-axis, this has the formula
, or
; this is the formula of a circle with a radius of
, and thus has an area of
. To find the volume, we integrate this formula over z. The maximum and minimum values of z occur when
, and then
, thus
; these are our limits of integration. Thus, the volume is:




 

Problems:

P1. Evaluate
 for the following functions:
(a)

(b)

(c)


P2. Given that
,
, and
, solve the following:
(a)

(b)

(c)

(d)

(e)
, where


P3. Find
 for the following functions:
(a)

(b)


P4. Use the power rule of differentiation to differentiate the following:
(a)

(b)

(c)


P5. Calculate the area under the curve of the function
 on the interval [2,5] using:
(a) right-hand approximation with 6 subdivisions.
(b) left-hand approximation with 3 subdivisions.

P6. Find the position, velocity, and acceleration for the following at
:
(a) A train begins moving and its displacement (in meters) is:

(b) A car begins moving and its displacement (in feet) is:


P7. Find
 given the equation
.

P8. Use the first fundamental theorem of calculus to evaluate:
(a)

(b)

P9. Find the total distance traveled from
 to
 by an object moving with a velocity in meters per second given by the equation
.

Solutions:
P1. (a) 7 is a constant function, so therefore,

(b)
 can be simplified by factoring:

(c)
, when
 for
 and
 for





Therefore,
, does not

P2. (a)
, where k is a constant, so

(b)
  so

(c)
, so

(d)
, if
, so

(e)
, so


P3. Using the formula
:

(a)







(b)






P4. The power rule is useful for finding the derivative of polynomial functions. It states that the derivative of
.
(a) 
(b) 
(c) 

P5. (a) First, divide the interval [2,5] by the number of subdivisions,
, thus each rectangle has a width of 0.5.
The height of the right-hand side of each rectangle is given by the value of the function at the points
. Summing these heights and multiplying by the width of each rectangle gives the approximate total area under the curve.



The total area under the curve of the function
 on the interval [2,5], calculated using right-hand approximation, is found to be approximately 53.375.
(b) First, divide the interval [2,5] by the number of subdivisions,
, thus each rectangle has a width of 1.

The height of the left-hand side of each rectangle is given by the value of the function at , 3, 4. Summing these heights and multiplying by the width of each rectangle gives the approximate area under the curve.
the function
 on the interval [2,5], calculated using left-hand approximation, is found to be approximately 38.

P6. Find each derivative and evaluate at
:

(a)











(b)













P7. Take the derivative of each term with respect to
:
.

In order to take the derivative of the left side, we will need to use the product rule
 where
 and
:




Now we will use the chain rule
 where
 and
:










P8. (a) By the first fundamental theorem of calculus:







By the first fundamental theorem of calculus:








P9. This velocity function is zero when
 or 6.
 is positive when
 and negative when
. Thus, the