Area and Volume Calculations: Master Integration Techniques

Question 1

Find the length of the curve y = x^3/3 + 1/4x from x = 1 to x = 4.

Accepted Answer

3√(13) - 2

Answer

3√(13) + 2

Answer

√(13) - 2

Answer

√(13) + 2

Question 2

Determine the average value of the function f(x) = sin(x) on the interval [0, π].

Accepted Answer

1/2

Answer

0

Answer

π/2

Answer

1

Question 3

Evaluate the definite integral ∫(x^3 + 2x^2 - 5)dx between x = 1 and x = 3.

Accepted Answer

21

Answer

14

Answer

7

Answer

0

Question 4

Calculate the area of the surface generated by revolving the curve y = ln(x) from x = 1 to x = 2 about the x-axis.

Accepted Answer

2π(eln(2) - 1)

Answer

π(eln(2) - 1)

Answer

2π(1 - eln(2))

Answer

π(1 - eln(2))

Question 5

Which method is used to calculate the area of the region enclosed by the curves y = sin(x) and y = cos(x) from x = 0 to x = π/2?

Accepted Answer

Integration

Answer

Trapezoidal rule

Answer

Midpoint rule

Answer

Simpson's rule

Question 6

Evaluate the improper integral ∫[0,∞) e^(-x^2)dx.

Accepted Answer

√π/2

Answer

1/2

Answer

1

Answer

π

Question 7

Calculate the volume of a solid with a circular base of radius 'r' and height 'h', where the cross-sections perpendicular to the base are equilateral triangles.

Accepted Answer

(√3/4)πr²h

Answer

πr²h

Answer

2πr²h

Answer

(1/3)πr²h

Question 8

Find the volume of a solid sphere with a radius of 5 centimeters. Express your answer in terms of **π**.

Accepted Answer

(4/3)π(5)³ cm³

Answer

4π(5)² cm³

Answer

10π(5)³ cm³

Question 9

Which of the following is the correct formula for calculating the volume of the solid obtained by revolving the region bounded by the curves y = x and y = 2x about the y-axis?

Accepted Answer

π∫[0,1] (2x - x^2)^2 dx

Answer

π∫[0,1] (x^2 - 2x)^2 dx

Answer

2π∫[0,1] (2x - x^2) dx

Answer

π∫[0,1] (x - 2x^2) dx

Question 10

Calculate the center of mass of the region bounded by the curves y = x^2 and y = 4 with respect to the x-axis.

Accepted Answer

8/5

Answer

4/5

Answer

16/5

Answer

2/5

Question 11

Calculate the area of the surface generated by rotating the curve y = √x from x = 1 to x = 4 about the y-axis.

Accepted Answer

(4π/3)(√4 - √1)

Answer

2π(4 - 1)

Answer

2π(√4 - √1)

Answer

(4π/3)(4 - 1)

Question 12

Find the volume of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1).

Accepted Answer

1/6

Answer

1/2

Answer

1/3

Question 13

Calculate the volume of the solid generated by rotating the region bounded by the polar curve r = sin(2θ) about the y-axis.

Accepted Answer

π/4

Answer

2π

Answer

π/2

Answer

π/8

Question 14

A spherical balloon has a diameter of 10 inches. What is the volume of the balloon?

Accepted Answer

4/3π(5)^3 cubic inches

Answer

2/3π(5)^3 cubic inches

Answer

8/3π(5)^3 cubic inches

Answer

π(5)^3 cubic inches

Question 15

A cone has a radius of 4 cm and a height of 8 cm. What is the volume of the cone?

Accepted Answer

32π/3 cubic cm

Answer

64π/3 cubic cm

Answer

16π cubic cm

Question 16

Find the area of the surface of the solid generated by rotating the curve y = ln(x), from x = 1 to x = e, about the x-axis.

Accepted Answer

(π(e - 1))^2

Answer

(2π(e - 1))^2

Answer

(π(e - 1))/2

Answer

(π(e - 1))/4

Question 17

What is the volume of the solid formed when the region bounded by the curves *y = x* and *y = x²* is rotated about the *x*-axis?

Accepted Answer

π/6

Answer

π/3

Answer

π/4

Answer

π/2

Question 18

Calculate the area enclosed by the curve *y = sin(x)*, the *x*-axis, and the vertical lines *x = 0* and *x = π/2*.

Accepted Answer

1

Answer

2

Answer

0

Answer

-1

Question 19

Determine the surface area generated by rotating the curve *y = x²* from *x = 0* to *x = 2* about the *x*-axis.

Accepted Answer

(8π√5)/3

Answer

(4π√5)/3

Answer

(2π√5)/3

Answer

(16π√5)/3

Question 20

Find the area of the region enclosed by the curves *y = x² - 4* and *y = 2x*.

Accepted Answer

36/3

Answer

8

Answer

4/3

Answer

16/3

Question 21

Calculate the area of the region bounded by the graphs of *y = x²* and *y = 2x - x²*.

Accepted Answer

4/3

Answer

1/3

Answer

1

Answer

2/3

Question 22

What is the volume of the solid generated by rotating the region bounded by the curves *y = √x*, the *x*-axis, and the line *x = 4* about the *x*-axis?

Accepted Answer

8π

Answer

4π

Answer

16π

Answer

32π

Question 23

Find the volume of the solid formed when the region bounded by the curves *y = x²* and *y = 4* is rotated around the *y*-axis.

Accepted Answer

32π/5

Answer

8π/5

Answer

16π/5

Answer

64π/5

Question 24

The region bounded by the curves *y = sin(x)*, *y = cos(x)*, *x = 0*, and *x = π/4* is revolved about the *x*-axis. What is the volume of the resulting solid?

Accepted Answer

π(√2 - 1)/2

Answer

π(√2 + 1)

Answer

π(√2 + 1)/2

Answer

π(√2 - 1)

Question 25

A solid has a base in the shape of the region bounded by the curve *y = x²* and the *x*-axis from *x = 0* to *x = 2*.  Cross-sections perpendicular to the *x*-axis are squares. What is the volume of this solid?

Accepted Answer

32/3

Answer

64/3

Answer

16/3

Answer

8/3

Question 26

Find the area of the region bounded by the curve *y = ln(x)*, the *x*-axis, and the vertical lines *x = 1* and *x = e*.

Accepted Answer

1

Answer

e

Answer

2

Answer

e - 1

Question 27

A solid is generated by rotating the region bounded by the curves y = e^x, y = 0, x = 0, and x = 1 about the y-axis. What is the volume of this solid?

Accepted Answer

π(e² - 1)

Answer

π(e² + 1)

Answer

π(e + 1)

Answer

π(e - 1)

Question 28

Find the area of the region enclosed by the curves y = x³ and y = x.

Accepted Answer

1/2

Answer

3/2

Answer

1

Answer

2

Question 29

A solid has a base bounded by the circle x² + y² = 4. Cross-sections perpendicular to the x-axis are equilateral triangles. What is the volume of this solid?

Accepted Answer

16√3π/3

Answer

8√3π/3

Answer

4√3π/3

Answer

32√3π/3

Question 30

The region bounded by the curves y = x and y = x² is rotated about the line x = 2. Find the volume of the resulting solid.

Accepted Answer

π/6

Answer

π

Answer

π/3

Answer

π/2

Question 31

Find the area of the region bounded by the parabola y = x^2 and the line y = 4.

Accepted Answer

16/3

Answer

32

Answer

16

Answer

8/3

Question 32

Calculate the volume of the solid generated by rotating the region bounded by the curves y = x and y = x^3 about the x-axis.

Accepted Answer

π/10

Answer

π/2

Answer

2π

Answer

π

Question 33

Find the surface area of the solid of revolution generated by rotating the curve y = sin(x) from x = 0 to x = π about the x-axis.

Accepted Answer

4π

Answer

16π

Answer

2π

Answer

8π

Question 34

Compute the work done by a constant force of 20 Newtons in moving an object 5 meters along a straight line.

Accepted Answer

100 Joules

Answer

200 Joules

Answer

50 Joules

Answer

150 Joules

Question 35

Find the centroid of the region bounded by the parabola y = x^2 and the line y = 4.

Accepted Answer

(8/5, 16/5)

Answer

(2, 6)

Answer

(4, 8)

Question 36

Compute the moment of inertia of the solid generated by rotating the region bounded by the curves y = x and y = x^3 about the y-axis.

Accepted Answer

2π/9

Answer

8π/9

Answer

4π/9

Answer

π/9

Question 37

Find the volume of the solid bounded by the cylinder x^2 + y^2 = 4 and the planes z = 0 and z = 3.

Accepted Answer

12π

Answer

18π

Answer

24π

Answer

6π

Question 38

Compute the center of mass of the region bounded by the parabolas y = x^2 and y = 2x.

Accepted Answer

(3/5, 8/5)

Answer

(2, 4)

Answer

(1, 2)

Question 39

Find the surface area of the solid generated by rotating the curve y = e^x from x = 0 to x = 1 about the y-axis.

Accepted Answer

π(e - 1)

Answer

2π(e + 1)

Answer

π(e + 1)

Answer

2π(e - 1)

Question 40

Compute the volume of the solid generated by rotating the region bounded by the curves y = √x and y = x^2 about the line y = 2.

Accepted Answer

16π/15

Answer

8π/15

Answer

32π/15

Answer

64π/15

Question 41

Find the area of the region bounded by the curves y = x² and y = 2x - x².

Accepted Answer

1

Answer

2/3

Answer

2

Answer

4/3

Question 42

Find the volume of the solid generated by revolving the region bounded by the curves y = √x, y = 0, and x = 4 about the x-axis.

Accepted Answer

8π

Answer

16π

Answer

32π

Answer

4π

Question 43

Find the volume of the solid generated by revolving the region bounded by the curves y = x² and y = x about the y-axis.

Accepted Answer

π/6

Answer

π/3

Answer

π/2

Question 44

Find the arc length of the curve y = (1/3)x^(3/2) from x = 0 to x = 4.

Accepted Answer

8/3

Answer

16/3

Answer

32/3

Answer

4/3

Question 45

Find the surface area of the solid generated by revolving the curve y = √x from x = 0 to x = 4 about the x-axis.

Accepted Answer

(8π/3)(5√5 - 1)

Answer

(16π/3)(5√5 - 1)

Answer

(32π/3)(5√5 - 1)

Answer

(4π/3)(5√5 - 1)

Question 46

The region bounded by the curves y = x² and y = x is revolved about the line x = 2. Find the volume of the resulting solid.

Accepted Answer

π/6

Answer

π

Answer

π/2

Answer

π/3

Question 47

Find the work done in pumping all the water out of a cylindrical tank that is 10 feet tall and has a radius of 5 feet, if the water is pumped to a height of 15 feet above the top of the tank.

Accepted Answer

156,250π ft-lb

Answer

625,000π ft-lb

Answer

312,500π ft-lb

Answer

78,125π ft-lb

Question 48

Find the hydrostatic force on a rectangular plate submerged vertically in water, with its top edge 2 feet below the surface and its bottom edge 8 feet below the surface, if the plate is 4 feet wide.

Accepted Answer

12480 lb

Answer

24960 lb

Answer

49920 lb

Answer

6240 lb

Question 49

Find the area of the region bounded by the curves y = sin(x), y = cos(x), x = 0, and x = π/4.

Accepted Answer

√2 - 1

Answer

1 - √2

Answer

√2 + 1

Question 50

Find the volume of the solid generated by revolving the region bounded by the curves y = e^x, y = 0, x = 0, and x = 1 about the x-axis.

Accepted Answer

π(e^2 - 1)/2

Answer

π(e - 1)/2

Answer

π(e + 1)/2

Answer

π(e^2 + 1)/2