What does the double integral $\iint_R f(x,y)\,dA$ represent when $f(x,y) \geq 0$?

The volume under $z=f(x,y)$ over $R$

The perimeter of the region $R$

The length of the boundary of $R$

Which integral is obtained by reversing the order of $\int_0^2\int_0^3 f(x,y)\,dy\,dx$?

$\int_0^3\int_0^2 f(x,y)\,dx\,dy$

$\int_0^2\int_0^3 f(x,y)\,dx\,dy$

$\int_0^3\int_0^2 f(x,y)\,dy\,dx$

$\int_0^2\int_0^3 f(x,y)\,dy\,dx$

Reverse the order of integration for $\int_0^1\int_x^1 f(x,y)\,dy\,dx$.

$\int_0^1\int_0^y f(x,y)\,dx\,dy$

$\int_0^1\int_y^1 f(x,y)\,dx\,dy$

$\int_0^1\int_0^x f(x,y)\,dy\,dx$

$\int_0^1\int_x^1 f(x,y)\,dx\,dy$

Reverse the order of integration for $\int_0^1\int_y^1 f(x,y)\,dx\,dy$.

$\int_0^1\int_0^x f(x,y)\,dy\,dx$

$\int_0^1\int_x^1 f(x,y)\,dy\,dx$

$\int_0^1\int_0^y f(x,y)\,dx\,dy$

$\int_0^1\int_y^1 f(x,y)\,dy\,dx$

What is the Jacobian of a transformation $x=x(u,v)$ and $y=y(u,v)$?

$\frac{\partial(x,y)}{\partial(u,v)}$

$\frac{\partial x}{\partial u}+\frac{\partial y}{\partial v}$

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}$

$\frac{\partial(x+y)}{\partial(u+v)}$

MTH165 Unit 5: Multivariable integration and applications - MCQ Practice | B. Tech. Aerospace Engineering

1 $What does the double integral represent when ?$

double integrals Easy

A.

The slope of over

B.

The perimeter of the region

C.

The length of the boundary of

D.

The volume under over

2 $Evaluate .$

double integrals Easy

A.

B.

C.

D.

3 $Evaluate .$

double integrals Easy

A.

B.

C.

D.

4 $In the iterated integral, which variable is integrated first?$

double integrals Easy

A.

B.

C.

D.

5 $Which integral is obtained by reversing the order of ?$

change of order of integration Easy

A.

B.

C.

D.

6 $Reverse the order of integration for .$

change of order of integration Easy

A.

B.

C.

D.

7 $Reverse the order of integration for .$

change of order of integration Easy

A.

B.

C.

D.

8 $Evaluate .$

triple integrals Easy

A.

B.

C.

D.

9 $Which expression is the Cartesian volume element in three dimensions?$

triple integrals Easy

A.

B.

C.

D.

10 $Evaluate .$

triple integrals Easy

A.

B.

C.

D.

11 $What is the Jacobian of a transformation and ?$

change of variables Easy

A.

B.

C.

D.

12 $Which equation correctly converts polar coordinates to Cartesian coordinates?$

change of variables Easy

A.

B.

C.

D.

13 $What is the area element in polar coordinates?$

change of variables Easy

A.

B.

C.

D.

14 $What is the volume element in cylindrical coordinates?$

change of variables Easy

A.

B.

C.

D.

15 $Which double integral gives the area of a plane region ?$

application of double integrals to calculate area and volume Easy

A.

B.

C.

D.

16 $What is the area of the unit circle found from ?$

application of double integrals to calculate area and volume Easy

A.

B.

C.

D.

17 $Which expression gives the volume under and above a region ?$

application of double integrals to calculate area and volume Easy

A.

B.

C.

D.

18 $A solid lies under the plane and above a rectangle of dimensions by . What is its volume?$

application of double integrals to calculate area and volume Easy

A.

B.

C.

D.

19 $Which triple integral gives the volume of a solid region ?$

application of triple integrals to calculate volume Easy

A.

B.

C.

D.

20 $Use a triple integral to find the volume of the unit cube .$

application of triple integrals to calculate volume Easy

A.

B.

C.

D.

21 $Evaluate .$

double integrals Medium

A.

B.

C.

D.

22 $Let be the triangular region bounded by,, and . Find .$

double integrals Medium

A.

B.

C.

D.

23 $Reverse the order of integration in .$

change of order of integration Medium

A.

B.

C.

D.

, because the lower boundary becomes the -axis

24 $Which integral represents after changing the order of integration?$

change of order of integration Medium

A.

B.

C.

D.

25 $Evaluate .$

triple integrals Medium

A.

B.

C.

D.

26 $Evaluate .$

triple integrals Medium

A.

B.

C.

D.

27 $Using polar coordinates, evaluate, where is the unit disk.$

change of variables Medium

A.

B.

C.

D.

28 $For the transformation and, what is ?$

change of variables Medium

A.

B.

C.

, since both inverse equations contain a factor of one-half

D.

29 $Use a double integral to determine the area enclosed by .$

application of double integrals to calculate area and volume Medium

A.

B.

C.

D.

30 $Find the volume below and above the triangular region,, and .$

application of double integrals to calculate area and volume Medium

A.

B.

C.

D.

31 $Using spherical coordinates, find the volume of the sphere .$

application of triple integrals to calculate volume Medium

A.

B.

C.

D.

, obtained by integrating only the radial factor and omitting the angular normalization

32 $Find the volume enclosed by the paraboloid and the plane .$

application of triple integrals to calculate volume Medium

A.

B.

C.

D.

33 $Evaluate, where .$

double integrals Medium

A.

B.

C.

D.

34 $Evaluate .$

double integrals Medium

A.

B.

C.

D.

35 $Evaluate by changing the order of integration.$

change of order of integration Medium

A.

B.

C.

D.

36 $Under the transformation and, a region maps to and . What is the area of the original region in the -plane?$

change of variables Medium

A.

, because the transformed rectangle has side lengths two and one before applying the inverse mapping

B.

C.

D.

37 $Evaluate, where is defined in cylindrical coordinates by,, and .$

triple integrals Medium

A.

B.

C.

D.

38 $Find the volume under the cone and above the -plane.$

application of double integrals to calculate area and volume Medium

A.

B.

C.

D.

39 $Find the volume of the first-octant solid bounded by and the coordinate planes.$

application of triple integrals to calculate volume Medium

A.

B.

C.

D.

40 $Use polar coordinates to find the area of the annular region .$

change of variables Medium

A.

B.

, found by subtracting the radii before squaring and then multiplying by the outer circumference

C.

D.

41 $Evaluate, where .$

double integrals Hard

A.

B.

C.

D.

42 $Let be the unit disk . Find .$

double integrals Hard

A.

B.

C.

D.

43 $Evaluate .$

double integrals Hard

A.

B.

C.

D.

44 $Which integral reverses the order in ?$

change of order of integration Hard

A.

B.

C.

D.

45 $Reverse the order of integration in .$

change of order of integration Hard

A.

, because both original curves act as upper boundaries

B.

C.

D.

46 $Evaluate by changing the order of integration.$

change of order of integration Hard

A.

B.

C.

D.

47 $Which integral is equivalent to after reversing the order?$

change of order of integration Hard

A.

B.

C.

D.

48 $Find, where .$

triple integrals Hard

A.

B.

C.

D.

49 $Evaluate, where .$

triple integrals Hard

A.

B.

C.

D.

50 $For the ball, evaluate .$

triple integrals Hard

A.

B.

C.

, obtained by replacing both squared Cartesian coordinates with their spherical averages

D.

51 $Let be the region between the paraboloid and the plane . Evaluate .$

triple integrals Hard

A.

B.

C.

D.

52 $Under and, a region maps to and . Find .$

change of variables Hard

A.

B.

C.

D.

53 $In the first quadrant, let satisfy and . Evaluate using and .$

change of variables Hard

A.

B.

C.

D.

54 $For, evaluate .$

change of variables Hard

A.

B.

C.

D.

55 $Let be bounded by,,, and . Evaluate .$

change of variables Hard

A.

B.

C.

D.

56 $Find the area inside the circle but outside the circle .$

application of double integrals to calculate area and volume Hard

A.

B.

C.

D.

57 $Find the volume enclosed between the paraboloids and .$

application of double integrals to calculate area and volume Hard

A.

B.

C.

D.

58 $For, determine the area of the astroid region .$

application of double integrals to calculate area and volume Hard

A.

B.

C.

D.

59 $Find the volume inside the sphere and above the cone .$

application of triple integrals to calculate volume Hard

A.

B.

C.

D.

60 $Find the volume common to the perpendicular cylinders and, where .$

application of triple integrals to calculate volume Hard

A.

B.

C.

, obtained by treating both perpendicular cylinders as rotationally symmetric about a common axis

D.

Unit 5: Multivariable integration and applications - Practice Quiz

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