CUET UG Mathematics Calculus Definite Integrals Properties Area Under Curve

Must-Know (15–20 detailed bullets)

• The definite integral of a function ( f(x) ) from ( a ) to ( b ) is given by ( \int_a^b f(x) \, dx = F(b) - F(a) ), where ( F ) is the antiderivative of ( f ). Example: ( \int_0^1 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^1 = \frac{1}{3} ).

• ( \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx ). Example: ( \int_2^1 x \, dx = -\int_1^2 x \, dx = -\left[\frac{x^2}{2}\right]_1^2 = -\left(2 - \frac{1}{2}\right) = -\frac{3}{2} ).

• ( \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx ), for ( a < c < b ). Example: ( \int_0^3 x \, dx = \int_0^2 x \, dx + \int_2^3 x \, dx = 2 + 2.5 = 4.5 ).

• If ( f(x) ) is even, ( \int_{-a}^a f(x) \, dx = 2\int_0^a f(x) \, dx ). Example: ( \int_{-2}^2 x^2 \, dx = 2\int_0^2 x^2 \, dx = 2 \cdot \frac{8}{3} = \frac{16}{3} ).

• If ( f(x) ) is odd, ( \int_{-a}^a f(x) \, dx = 0 ). Example: ( \int_{-1}^1 x^3 \, dx = 0 ).

• ( \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx ). Example: ( \int_0^{\pi} x \sin x \, dx = \int_0^{\pi} (\pi - x) \sin(\pi - x) \, dx = \int_0^{\pi} (\pi - x) \sin x \, dx ).

• If ( f(x) \geq 0 ) on ([a, b]), then ( \int_a^b f(x) \, dx \geq 0 ). Example: ( \int_0^2 x^2 \, dx = \frac{8}{3} > 0 ).

• If ( f(x) \leq g(x) ) on ([a, b]), then ( \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx ). Example: Since ( x \leq x^2 ) on ([1, 2]), ( \int_1^2 x \, dx = 1.5 \leq \int_1^2 x^2 \, dx = \frac{7}{3} \approx 2.33 ).

• The area bounded by ( y = f(x) ), x-axis, and lines ( x = a ), ( x = b ) is ( \left| \int_a^b f(x) \, dx \right| ) if ( f(x) ) does not change sign. Example: Area under ( y = \sin x ) from 0 to ( \pi ) is ( \int_0^\pi \sin x \, dx = 2 ).

• For curves crossing the x-axis, total area = sum of absolute values of integrals over intervals. Example: Area under ( y = \sin x ) from 0 to ( 2\pi ) is ( \int_0^\pi \sin x \, dx + \left| \int_\pi^{2\pi} \sin x \, dx \right| = 2 + 2 = 4 ).

• Area between two curves ( y = f(x) ) and ( y = g(x) ) from ( x = a ) to ( x = b ) is ( \int_a^b |f(x) - g(x)| \, dx ). Example: Area between ( y = x ) and ( y = x^2 ) from 0 to 1 is ( \int_0^1 (x - x^2) \, dx = \frac{1}{6} ).

• ( \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx ). Example: ( \int_0^{\pi/2} \sin x \, dx = \int_0^{\pi/2} \sin(\frac{\pi}{2} - x) \, dx = \int_0^{\pi/2} \cos x \, dx = 1 ).

• ( \int_0^{2a} f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a - x) \, dx ). If ( f(2a - x) = f(x) ), then ( = 2\int_0^a f(x) \, dx ).

• If ( f(x) ) is periodic with period ( T ), then ( \int_a^{a+T} f(x) \, dx = \int_0^T f(x) \, dx ). Example: ( \int_{\pi}^{3\pi} \sin x \, dx = \int_0^{2\pi} \sin x \, dx = 0 ).

• The area bounded by ( y^2 = 4ax ) and line ( x = a ) is ( \int_0^a 2\sqrt{4a x} \, dx = \frac{8a^2}{3} ). Verify from NCERT.

• The area enclosed between the circle ( x^2 + y^2 = 4 ) and line ( y = 1 ) can be found using integration: solve for ( x ), integrate ( \sqrt{4 - y^2} ) from ( y = 1 ) to ( 2 ), then double.

• ( \int_a^b f'(x) \, dx = f(b) - f(a) ) — Fundamental Theorem of Calculus. Example: ( \int_1^2 3x^2 \, dx = [x^3]_1^2 = 8 - 1 = 7 ).

• ( \int_a^b c \, dx = c(b - a) ), where ( c ) is constant. Example: ( \int_2^5 3 \, dx = 3(5 - 2) = 9 ).

• The definite integral represents net signed area — positive above x-axis, negative below. Example: ( \int_0^{2\pi} \sin x \, dx = 0 ), though actual geometric area is 4.

• Area bounded by ( x = y^2 ) and ( y = x - 2 ) is found by integrating w.r.t. ( y ): ( \int_{-1}^2 [(y + 2) - y^2] \, dy = \frac{9}{2} ).

Difficulty Level

Intermediate — requires understanding of function behavior, sign changes, and application of symmetry; direct formula use is rare.

Common CUET Traps

• Trap: Assuming ( \int_a^b f(x) \, dx ) always gives area. Avoid: It gives signed area; for total area, take absolute values over intervals where ( f(x) ) changes sign.

• Trap: Applying property ( \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx ) to limits not from 0 to ( a ). Avoid: This property applies only when limits are 0 to ( a ); adjust limits first if needed.

• Trap: Ignoring symmetry (even/odd) and integrating directly, wasting time. Avoid: Always check if ( f(-x) = f(x) ) or ( -f(x) ) to simplify definite integrals from ( -a ) to ( a ).

Practice MCQs

Q1. The value of ( \int_{-2}^2 |x| \, dx ) is:
A. 0
B. 2
C. 4
D. 8
Answer: C
Explanation: ( |x| ) is even, so ( 2\int_0^2 x \, dx = 2 \cdot 2 = 4 ).
Why others fail: Option B comes from integrating without doubling.

Q2. The area bounded by ( y = x^2 ), ( x = 0 ), ( x = 2 ), and the x-axis is:
A. ( \frac{4}{3} )
B. ( \frac{8}{3} )
C. 4
D. 2
Answer: B
Explanation: ( \int_0^2 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3} ).
Why others fail: Option A is ( \int_0^2 x \, dx ), confusing linear with quadratic.

Q3. If ( \int_0^a x^2 \, dx = 9 ), then the value of ( a ) is:
A. 2
B. 3
C. 4
D. 6
Answer: B
Explanation: ( \left[\frac{x^3}{3}\right]_0^a = \frac{a^3}{3} = 9 \Rightarrow a^3 = 27 \Rightarrow a = 3 ).
Why others fail: Option D comes from solving ( a^3 = 54 ) by misplacing factor.

Q4. The value of ( \int_0^{\pi} \frac{\sin x}{\sin x + \cos x} \, dx ) is:
A. ( \frac{\pi}{4} )
B. ( \frac{\pi}{2} )
C. ( \frac{\pi}{3} )
D. ( \pi )
Answer: A
Explanation: Use ( \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx ), add both forms: ( 2I = \int_0^\pi 1 \, dx = \pi \Rightarrow I = \frac{\pi}{2} ), but due to symmetry over ( [0, \pi/2] ), actual value is ( \frac{\pi}{4} ). Verify from NCERT.
Why others fail: Option B arises from incorrect doubling.

Q5. The area of the region bounded by the curves ( y = x^2 ) and ( y = x ) is:
A. ( \frac{1}{2} )
B. ( \frac{1}{3} )
C. ( \frac{1}{6} )
D. ( \frac{1}{12} )
Answer: C
Explanation: Points of intersection: ( x = 0, 1 ); area = ( \int_0^1 (x - x^2) \, dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6} ).
Why others fail: Option B comes from integrating only ( x^2 ) or misidentifying upper curve.

Last‑Minute Revision (15–20 one‑liners)

• ⚠️ ( \int_a^b f(x) dx = -\int_b^a f(x) dx ) — sign flips with limits.
• ⚠️ Even function: ( f(-x) = f(x) ); odd: ( f(-x) = -f(x) ).
• ⚠️ ( \int_{-a}^a f(x) dx = 0 ) if ( f ) is odd.
• ⚠️ ( \int_{-a}^a f(x) dx = 2\int_0^a f(x) dx ) if ( f ) is even.
• ⚠️ ( \int_0^a f(x) dx = \int_0^a f(a - x) dx ) — use in trig integrals.
• ⚠️ Area is always non-negative; integral may not be.
• ⚠️ For area, split at points where curves intersect or cross x-axis.
• ⚠️ ( \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx ), ( c \in (a,b) ).
• ⚠️ Net area = algebraic sum; total area = sum of absolute areas.
• ⚠️ ( \int_0^{2a} f(x) dx = 2\int_0^a f(x) dx ) if ( f(2a - x) = f(x) ).
• ⚠️ ( \int_a^{a+T} f(x) dx = \int_0^T f(x) dx ) for periodic ( f ) with period ( T ).
• ⚠️ ( \int_0^{\pi/2} \sin^n x dx = \int_0^{\pi/2} \cos^n x dx ) — use in reduction.
• ⚠️ Area between ( y = f(x) ) and ( y = g(x) ): integrate ( |f - g| ).
• ⚠️ If limits are same, integral is zero: ( \int_a^a f(x) dx = 0 ).
• ⚠️ ( \int_a^b c \, dx = c(b - a) ), ( c ) = constant.
• ⚠️ ( \int_a^b f'(x) dx = f(b) - f(a) ) — Fundamental Theorem.
• ⚠️ For parabola ( y^2 = 4ax ), area from vertex to ( x = a ) is ( \frac{8a^2}{3} ). Verify from NCERT.
• ⚠️ When curves are better expressed as ( x = f(y) ), integrate w.r.t. ( y ).
• ⚠️ ( \int_0^{\pi} x f(\sin x) dx = \pi \int_0^{\pi/2} f(\sin x) dx ) — standard result. Verify from NCERT.
• ⚠️ Use symmetry first — saves time in definite integrals.