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Unit 5: Multivariable integration and applications - Practice Quiz

MTH165 — Mathematics For Engineers 60 Questions
0 Correct 0 Wrong 60 Left
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1 What does the double integral represent when ?

Double integrals Easy
A. The perimeter of the region
B. The length of a curve in
C. The volume under the surface over
D. The slope of the surface over

2 Evaluate .

Double integrals Easy
A.
B.
C.
D.

3 In the iterated integral , which variable is integrated first?

Double integrals Easy
A. Neither variable
B.
C.
D. Both simultaneously

4 Evaluate .

Double integrals Easy
A.
B.
C.
D.

5 Which integral is obtained by reversing the order of integration in ?

Change of order of integration Easy
A.
B.
C.
D.

6 The integral describes which region?

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 A triple integral is generally used to integrate a function over what kind of region?

Triple integrals Easy
A. A one-dimensional interval
B. A single boundary point
C. A two-dimensional curve
D. A three-dimensional region

9 Evaluate .

Triple integrals Easy
A.
B.
C.
D.

10 In , which variable is integrated first?

Triple integrals Easy
A.
B. All three together
C.
D.

11 Which coordinate transformation is used for polar coordinates?

Change of variables Easy
A.
B.
C.
D.

12 Under the polar-coordinate transformation, what does the area element become?

Change of variables Easy
A.
B.
C.
D.

13 What is the main purpose of the Jacobian in a change of variables?

Change of variables Easy
A. To adjust the area or volume element
B. To differentiate the final answer
C. To determine the function's maximum
D. To reverse the integration limits

14 Which double integral gives the area of a region ?

Application of double integrals to calculate area and volume Easy
A.
B.
C.
D.

15 Find the area of the rectangle , using a double integral.

Application of double integrals to calculate area and volume Easy
A. square units
B. square units
C. square units
D. square units

16 Which expression gives the volume under and above a region ?

Application of double integrals to calculate area and volume Easy
A.
B.
C.
D.

17 Find the volume under the plane above the unit square , .

Application of double integrals to calculate area and volume Easy
A. cubic units
B. cubic units
C. cubic units
D. cubic unit

18 Which triple integral gives the volume of a solid region ?

Application of triple integrals to calculate volume Easy
A.
B.
C.
D.

19 Find the volume of the box , , .

Application of triple integrals to calculate volume Easy
A. cubic units
B. cubic units
C. cubic units
D. cubic units

20 Evaluate the volume integral .

Application of triple integrals to calculate volume Easy
A. cubic units
B. cubic units
C. cubic unit
D. cubic units

21 Evaluate .

Double integrals Medium
A.
B.
C.
D.

22 Evaluate , where is the unit disk .

Double integrals Medium
A.
B.
C.
D.

23 Evaluate .

Double integrals Medium
A.
B.
C.
D.

24 Which expression represents the reversed order of integration for ?

Change of order of integration Medium
A.
B.
C.
D.

25 Reverse the order of integration in .

Change of order of integration Medium
A.
B.
C.
D.

26 After reversing the order, evaluate .

Change of order of integration Medium
A.
B.
C.
D.

27 Evaluate , where is the unit cube .

Triple integrals Medium
A.
B.
C.
D.

28 Which cylindrical-coordinate integral represents the volume of the cylinder , ?

Triple integrals Medium
A.
B.
C.
D.

29 Evaluate , where is the first-octant tetrahedron bounded by and the coordinate planes.

Triple integrals Medium
A.
B.
C.
D.

30 For the transformation and , what is the value of ?

Change of variables Medium
A.
B.
C.
D.

31 Under the transformation , , what is in terms of ?

Change of variables Medium
A.
B.
C.
D.

32 Use and to evaluate , where is bounded by , , , and .

Change of variables Medium
A.
B.
C.
D.

33 Find the area enclosed between the curves and .

Application of double integrals to calculate area and volume Medium
A.
B.
C.
D.

34 Find the volume under and above the disk .

Application of double integrals to calculate area and volume Medium
A.
B.
C.
D.

35 What is the area of the ellipse ?

Application of double integrals to calculate area and volume Medium
A.
B.
C.
D.

36 Find the volume of the sphere .

Application of triple integrals to calculate volume Medium
A.
B.
C.
D.

37 Find the volume of the first-octant tetrahedron bounded by and the coordinate planes.

Application of triple integrals to calculate volume Medium
A.
B.
C.
D.

38 Find the volume enclosed between the paraboloid and the plane .

Application of triple integrals to calculate volume Medium
A.
B.
C.
D.

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

Application of triple integrals to calculate volume Medium
A.
B.
C.
D.

40 Find the volume inside the cylinder between the planes and .

Application of triple integrals to calculate volume Medium
A.
B.
C.
D.

41 Evaluate the iterated integral

Double integrals Hard
A.
B.
C.
D.

42 Reverse the order of integration in

Change of order of integration Hard
A.
B.
C.
D.

43 Evaluate the improper double integral

Double integrals Hard
A.
B.
C.
D.

44 Let . Evaluate

Triple integrals Hard
A.
B.
C.
D.

45 For the solid ball , evaluate

Triple integrals Hard
A.
B.
C.
D.

46 Let . Evaluate using and .

Change of variables Hard
A.
B.
C.
D.

47 Let . Under , with , evaluate the improper integral

Change of variables Hard
A.
B.
C. The integral diverges at the origin
D.

48 For , find the area enclosed by the astroid

Application of double integrals to calculate area and volume Hard
A.
B.
C.
D.

49 For , determine the area common to the disks and .

Application of double integrals to calculate area and volume Hard
A.
B.
C.
D.

50 Find the volume enclosed between the paraboloids

Application of double integrals to calculate area and volume Hard
A.
B.
C.
D.

51 For the ellipsoid , evaluate

Triple integrals Hard
A.
B.
C.
D.

52 Find the volume common to the perpendicular cylinders

Application of triple integrals to calculate volume Hard
A.
B.
C.
D.

53 Evaluate by changing the order of integration.

Change of order of integration Hard
A.
B.
C.
D.

54 Which expression correctly reverses the order in

Change of order of integration Hard
A.
B.
C.
D.

55 For real , determine when the improper integral converges.

Double integrals Hard
A. It converges exactly when
B. It converges exactly when
C. It converges exactly when
D. It converges exactly when

56 For real , determine when the improper integral converges.

Triple integrals Hard
A. It converges exactly when
B. It converges exactly when
C. It converges exactly when
D. It converges exactly when

57 For , find the volume inside the sphere and inside the cone , where is measured from the positive -axis.

Application of triple integrals to calculate volume Hard
A.
B.
C.
D.

58 In the first quadrant, let Find the area of using and .

Change of variables Hard
A.
B.
C.
D.

59 Evaluate the improper integral

Double integrals Hard
A. The integral diverges because at the corner
B.
C.
D.

60 Find the volume of the solid

Application of triple integrals to calculate volume Hard
A.
B.
C.
D.
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