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Unit 6: Integral Calculus - Practice Quiz

MTH174 — Engineering Mathematics 60 Questions
0 Correct 0 Wrong 60 Left
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1 A double integral is generally used to integrate a function over which type of region?

Double integrals Easy
A. A plane region
B. A time interval
C. A single point
D. A line segment

2 What does the integral represent?

Double integrals Easy
A. The slope of
B. The volume of
C. The perimeter of
D. The area of

3 Which notation represents a double integral over a region ?

Double integrals Easy
A.
B.
C.
D.

4 Evaluate .

Double integrals Easy
A.
B.
C.
D.

5 Changing the order of integration means changing the order of which variables are integrated?

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

6 The integral describes the region satisfying which inequalities?

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

7 When changing the order of integration, what must remain unchanged?

Change of order of integration Easy
A. The integration region
B. The inner variable
C. The differential symbols
D. The outer limits

8 Which order is the reverse of ?

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

9 A change of variables replaces the original variables with which type of variables?

Change of variables Easy
A. Integration limits
B. Constant values
C. New variables
D. Function values

10 In a two-dimensional change of variables, the Jacobian is used to change which element?

Change of variables Easy
A. The integration sign
B. The coordinate labels
C. The area element
D. The function name

11 In polar coordinates, the Cartesian area element becomes which expression?

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

12 Which coordinates are commonly used to describe circular regions?

Change of variables Easy
A. Matrix coordinates
B. Polar coordinates
C. Sequence coordinates
D. Linear coordinates

13 The volume under over a region is represented by which integral?

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

14 If over a region , what does calculate?

Application of double integrals to calculate area and volume Easy
A. The area of
B. The perimeter of
C. The average slope of
D. The maximum height of

15 What quantity is calculated by when ?

Application of double integrals to calculate area and volume Easy
A. Length between two points
B. Average value of one surface
C. Volume between two surfaces
D. Area of the boundary curve

16 A triple integral is used to integrate over which type of region?

Triple integrals Easy
A. A three-dimensional solid
B. A one-dimensional interval
C. A single coordinate
D. A two-dimensional curve

17 Which differential is commonly used in a triple integral in Cartesian coordinates?

Triple integrals Easy
A.
B.
C.
D.

18 Which notation represents a triple integral over a solid ?

Triple integrals Easy
A.
B.
C.
D.

19 The volume of a solid can be calculated using which triple integral?

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

20 Evaluate .

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

21 Evaluate .

Double integrals Medium
A.
B.
C.
D.

22 Find , where .

Double integrals Medium
A.
B.
C.
D.

23 Evaluate .

Double integrals Medium
A.
B.
C.
D.

24 Reverse the order of integration in .

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

25 Rewrite by changing the order of integration.

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

26 Evaluate by changing the order of integration.

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

27 Under the transformation and , find the absolute value of the Jacobian .

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

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

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

29 Using polar coordinates, evaluate , where is the annulus .

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

30 Find the area enclosed by and .

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

31 Find the volume under and above the triangular region , , .

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

32 Find the area of the ellipse using a suitable change of variables.

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

33 Find the volume under and above the unit disk .

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

34 Evaluate over the unit cube .

Triple integrals Medium
A.
B.
C.
D.

35 Evaluate , where is given by , , and .

Triple integrals Medium
A.
B.
C.
D.

36 In cylindrical coordinates, which integral represents the volume inside and between and ?

Triple integrals Medium
A.
B.
C.
D.

37 For the transformation , , and , what is the volume element?

Triple integrals Medium
A.
B.
C.
D.

38 Find the volume of the tetrahedron in the first octant bounded by .

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

39 Find the volume of the sphere using spherical coordinates.

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

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

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

41 Evaluate , where is the disk .

Double integrals Hard
A.
B.
C.
D.

42 Find .

Double integrals Hard
A.
B.
C.
D.

43 Rewrite by reversing the order of integration.

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

44 Reverse the order of integration for the region between and , with .

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

45 Let and . Evaluate , where maps to , .

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

46 Under and , with and , evaluate .

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

47 Find the area enclosed by the polar curve .

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

48 Find the volume under and above the -plane.

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

49 Evaluate , where is the annulus .

Double integrals Hard
A.
B.
C.
D.

50 Evaluate .

Triple integrals Hard
A.
B.
C.
D.

51 Evaluate .

Triple integrals Hard
A.
B.
C.
D.

52 Find the volume of the tetrahedron and , where .

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

53 Find the volume inside , above , and below .

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

54 Let , , and . If the transformed region is the unit cube , find .

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

55 Find the volume of the part of the sphere lying in the first octant and satisfying .

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

56 Evaluate , where .

Triple integrals Hard
A.
B.
C.
D.

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

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

58 Find the volume of the first-octant region satisfying .

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

59 Evaluate by changing the order of integration.

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

60 Find the area enclosed by the cardioid .

Application of double integrals to calculate area and volume Hard
A.
B.
C.
D.
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Unit 5