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Engineering Math Practice Test: Multiple Integrals
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Multiple Integrals topics include: Double integrals and its applications, variables changing in double and triple integrals, dirichlet’s integral, triple integral and its applications. Multiple integrals can involve two or more variables.  Multiple integrals are used in engineering, particularly in structures and mechanics, to determine the properties of plane and solid bodies. These properties include volume, mass, center of gravity, and moment of inertia.  Here are some types of multiple integrals: Double integral Used to find the surface area of a 2D figure. It can also be used to... Show more
Engineering Math Practice Test: Multiple Integrals
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25 Questions

1. Find the value of ∫∫xy7 Cos(x)Cos(y) dxdy.
2. Volume of an object expressed in spherical coordinates is given by \(V = ∫_0^2π∫_0^\frac{π}{3}∫_0^1 r cos∅ \,dr \,d∅ \,dθ.\) The value of the integral is _______
3. The integral value of \(\int_0^a \int_0^x \int_0^{x+y} e^{x+y+z} \,dz \,dy \,dx\) is given by _____
4. Find the value of integral \(\int_0^1\int_{x^2}^x xy(x+y)dydx\).
5. What is the value of integral \(∭_Re^{{(x^2+y^2+z^2)}^{\frac{3}{2}}} \,dx \,dy \,dz \) where R is the region given by x2+y2+z2≤1?
6. If ∭R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x=0, y=0, z=0, z=1 & x2+y2=1 then what is the value of that integral?
7. If double integral in Cartesian coordinate is given by ∬R f(x,y) dx dy then the value of same integral in polar form is _____
8. Which of the following equation represents Moment of Inertia of a plane region relative to x-axis?
9. The half-interval method in numerical analysis is also known as __________
10. Find the distance travelled by a car moving with acceleration given by a(t)=Sin(t), if it moves from t = 0 sec to t = π/2 sec, if velocity of a car at t=0sec is 10 km/hr.
11. Find the distance travelled by a car moving with acceleration given by a(t)=t2 – t, if it moves from t = 0 sec to t = 1 sec, if velocity of a car at t = 0sec is 10 km/hr.
12. Find the value of \(\int_0^{1-y} xy\sqrt{1-x-y} \,dxdy\) where, y varies from 0 to 1.
13. Evaluate ∫∫∫ 12y-8x dV in the region behind y=10-2z and bounded by z=2x, z=5 and x=0.
14. The volume of the region R defined by inequalities 0≤z≤1, 0≤y+z≤2,0≤x+y+z≤3 is given by ______
15. Assume a planet having a radius R and its density is expressed as = \(\frac{R+r}{2r}D’\).
16. Find the integration of \(\int\int_0^{\sqrt{2ax-x^2}}x \,dxdx\).
17. The integral of \(\int_{-1}^1 \int_0^z \int_{x-z}^{x+z} (x+y+z)\,dy \,dx \,dz\) is given by _______
18. What is the result of the integration \(∫_3^4∫_1^2(x^2+y)dxdy\)?
19. Find the integration of \(\int\int0x (x2 + y2) \,dxdy\), where x varies from 0 to 1.
20. The region bounded by circle is an example of regular domain.
21. Find the value of ∫∫ x⁄x2 + y2 dxdy.
22. Evaluate ∫∫[x2 + y2 – a2 ]dxdy where, x and y varies from –a to a.
23. Using change of variables principle in double integral we can reduce cartesian integral to simpler form.
24. The integral value of \(\int_0^1 \int_0^{1-x} \int_0^{1-x-y} \frac{dz dy dx}{(1+x+y+z)^3} \) is given by_____
25. Find the area inside a ellipse of minor-radius ‘b’ and major-radius ‘a’.

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