Deck 6: Applications of Integration

Question 1

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1,5].

-Using four disks or washers approximate the volume of the solid that is obtained by revolving this region around the x-axis.

A) 6 $\pi$
B) 7 $\pi$
C) 8 $\pi$
D) 9 $\pi$

Answer 1

8 $\pi$

Question 2

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1,5].

-Using four disks or washers approximate the volume of the solid that is obtained by revolving this region around the y-axis.

A) 33.27 $\pi$
B) 46.27 $\pi$
C) 53.27 $\pi$
D) 36.27 $\pi$

Answer 2

36.27 $\pi$

Question 3

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1,5].

-Using four disks or washers approximate the volume of the solid that is obtained by revolving this region around the vertical line x = 5.

A) 15.07 $\pi$
B) 17.07 $\pi$
C) 14.07 $\pi$
D) 12.07 $\pi$

Answer 3

17.07 $\pi$

Question 4

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1,5].

-Using four disks or washers approximate the volume of the solid that is obtained by revolving this region around the horizontal line y = -1.

A) 18.77 $\pi$
B) 28.77 $\pi$
C) 22.77 $\pi$
D) 15.77 $\pi$

Question 5

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2].

-Use disk /washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the x-axis.

A) 15.73 $\pi$
B) 19.73 $\pi$
C) 13.73 $\pi$
D) 11.73 $\pi$

Question 6

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

A) 10 $\pi$
B) 14 $\pi$
C) 16 $\pi$
D) 12 $\pi$

Question 7

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = -1.

A) 19.33 $\pi$
B) 16.33 $\pi$
C) 21.33 $\pi$
D) 23.33 $\pi$

Question 8

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 2.

A) 4.66 $\pi$
B) 6.66 $\pi$
C) 8.66 $\pi$
D) 9.66 $\pi$

Question 9

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = -1.

A) 20.07 $\pi$
B) 21.07 $\pi$
C) 23.07 $\pi$
D) 18.07 $\pi$

Question 10

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = 5.

A) 23.93 $\pi$
B) 32.93 $\pi$
C) 27.93 $\pi$
D) 29.93 $\pi$

Question 11

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

A) 2 $\pi$
B) 2.5 $\pi$
C) 3 $\pi$
D) 1.5 $\pi$

Question 12

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the x-axis.

A) 4.37 $\pi$
B) 3.47 $\pi$
C) 2.47 $\pi$
D) 5.47 $\pi$

Question 13

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 1.

A) 1.16 $\pi$
B) 2.16 $\pi$
C) 1.96 $\pi$
D) 2.67 $\pi$

Question 14

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 2.

A) 2.38 $\pi$
B) 2.83 $\pi$
C) 3.83 $\pi$
D) 4.83 $\pi$

Question 15

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = 2.

A) 1.87 $\pi$
B) 2.87 $\pi$
C) 3.87 $\pi$
D) 0.87 $\pi$

Question 16

Consider the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = 3 x$ on the interval [0, 3].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

A) 11.5 $\pi$
B) 12.5 $\pi$
C) 13.5 $\pi$
D) 14.5 $\pi$

Question 17

Consider the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = 3 x$ on the interval [0, 3].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the x-axis.

A) 42.6 $\pi$
B) 32.4 $\pi$
C) 24.7 $\pi$
D) 38.6 $\pi$

Question 18

Consider the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = 3 x$ on the interval [0, 3].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = -1.

A) 44.4 $\pi$
B) 34.4 $\pi$
C) 41.4 $\pi$
D) 24.1 $\pi$

Question 19

Consider the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = 3 x$ on the interval [0, 3].

-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 4.

A) 12.5 $\pi$
B) 20.5 $\pi$
C) 18.5 $\pi$
D) 22.5 $\pi$

Question 20

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1, 5].

-Using four shells approximate the volume of the solid that is obtained by revolving this region around the x-axis.

A) 9.25 $\pi$
B) 8.25 $\pi$
C) 10.25 $\pi$
D) 7.25 $\pi$

Question 21

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1, 5].

-Using four shells approximate the volume of the solid that is obtained by revolving this region around the y-axis.

A) 32.15 $\pi$
B) 30.15 $\pi$
C) 36.15 $\pi$
D) 41.15 $\pi$

Question 22

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1, 5].

-Using four shells approximate the volume of the solid that is obtained by revolving this region around the vertical line x = 5.

A) 15.68 $\pi$
B) 17.68 $\pi$
C) 13.68 $\pi$
D) 19.68 $\pi$

Question 23

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

A) $\pi$
B) 2 $\pi$
C) 1.5 $\pi$
D) 2.5 $\pi$

Question 24

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the x-axis.

A) 2.47 $\pi$
B) 3.47 $\pi$
C) 1.47 $\pi$
D) 4.47 $\pi$

Question 25

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 1.

A) 1.16 $\pi$
B) 2.16 $\pi$
C) 1.96 $\pi$
D) 2.67 $\pi$

Question 26

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].

-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 2.

A) 2.38 $\pi$
B) 2.83 $\pi$
C) 3.83 $\pi$
D) 4.83 $\pi$

Question 27

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2].

-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

A) 14.66 $\pi$
B) 10.66 $\pi$
C) 8.66 $\pi$
D) 15.66 $\pi$

Question 28

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2].

-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the x-axis.

A) 13.67 $\pi$
B) 15.67 $\pi$
C) 18.67 $\pi$
D) 11.67 $\pi$

Question 29

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2].

-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 2.

A) 10.33 $\pi$
B) 11.33 $\pi$
C) 12.33 $\pi$
D) 13.33 $\pi$

Question 30

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2].

-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = 4.

A) 20.33 $\pi$
B) 25.33 $\pi$
C) 29.33 $\pi$
D) 21.33 $\pi$

Question 31

Find the exact value of the arc length of the function $f ( x ) = 3 - x$ on the interval [1, 4] using a definite integral.

A) $4 \sqrt { 2 }$
B) $3 \sqrt { 2 }$
C) $5 \sqrt { 2 }$
D) $4 \sqrt { 3 }$

Question 32

Find the exact value of the arc length of the function $f ( x ) = 2 x + 3$ on the interval [-2, 3] using a definite integral.

A) $3 \sqrt { 5 }$
B) $4 \sqrt { 5 }$
C) $5 \sqrt { 5 }$
D) $5 \sqrt { 4 }$

Question 33

Find the exact value of the arc length of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval [-2, 2] using a definite integral.

Question 34

Find the exact value of the arc length of the function $f ( x ) = 2 x ^ { 3 / 2 }$ on the interval [0, 2] using a definite integral.

Question 35

Find the exact value of the arc length of the function $f ( x ) = \ln ( \sin x )$ on the interval [ $\pi$ /4, $\pi$ /2] using a definite integral.

Question 36

Find the exact value of the arc length of the function $f ( x ) = x ^ { 2 }$ on the interval [-1, 2] using a definite integral.

Question 37

Find the exact value of the arc length of the function $f ( x ) = 3 \cos x$ on the interval [0, $\pi$ ] using a definite integral.

Question 38

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = 2 x$ around the x-axis on the interval [0, 2].

A) $8 \sqrt { 5 } \pi$
B) $5 \sqrt { 8 } \pi$
C) $8 \sqrt { 3 } \pi$
D) $6 \sqrt { 5 } \pi$

Question 39

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = \frac { 1 } { x }$ around the x-axis on the interval [1, 4].

A) 2.33 $\pi$
B) 3.99 $\pi$
C) 2.99 $\pi$
D) 3.33 $\pi$

Question 40

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = e ^ { x }$ around the x-axis on the interval [-1, 1].

A) 4.45 $\pi$
B) 8.85 $\pi$
C) 6.85 $\pi$
D) 7.65 $\pi$

Question 41

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = e ^ { - x }$ around the x-axis on the interval [-1, 2].

A) 6.33 $\pi$
B) 8.33 $\pi$
C) 9.33 $\pi$
D) 10.33 $\pi$

Question 42

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = \sin x$ around the x-axis on the interval [ $\pi$ , 2 $\pi$ ].

A) 5.59 $\pi$
B) 4.59 $\pi$
C) 6.59 $\pi$
D) 3.59 $\pi$

Question 43

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = \cos x$ around the x-axis on the interval [ $\pi$ /2, 3 $\pi$ /2].

A) 5.59 $\pi$
B) 6.59 $\pi$
C) 3.59 $\pi$
D) 4.59 $\pi$

Question 44

Find the mass of a cylindrical rod with a radius of 5 centimeters and a length of 30 centimeters, made of two metals in such a way that the density of the rod × centimeters from the left end is $\rho ( x ) = 10 - 0.05 x$ grams per cubic centimeter.

A) 6525 $\pi$
B) 6975.5 $\pi$
C) 6937.5 $\pi$
D) 6637.5 $\pi$

Question 45

Find the mass of a 20-inch rod whose cross section is a 2 × 2 inch square, with density × inches from the left end given by $\rho ( x ) = 3.6 + 0.6 x - 0.03 x ^ { 2 }$ grams per cubic inch.

A) 440
B) 224
C) 446
D) 448

Question 46

Find the work required to pump all of the water out of the top of an upright conical tank with top radius 4 feet and height 6 feet.

Question 47

A cone-shaped water tank with top radius 4 feet and height 6 feet is on an 8-ft-high platform. Find the work done to fill this depot completely through an opening at the bottom of the tank if we pump the water from the ground level.

Question 48

Find the work required to pump the upper 3 feet of the water out of the top of an upright conical tank with top radius 4 feet and height 6 feet.

Question 49

Find the work required to pump all of the water out of the top of an upright cylindrical tank with radius 4 feet and height 8 feet.

Question 50

Find the work required to pump all of the water out of the top of a tank and up to the ground level, given that the tank is an upright cylinder with radius 4 feet and height 8 feet, buried so that its top is 2 feet below the surface.

Question 51

Find the hydrostatic force exerted on one of the long sides of a rectangular water tank that is 6 feet wide, 10 feet long, and 4 feet deep.

Question 52

Find the hydrostatic force exerted on a dam in the shape of a trapezoid whose top is 320 feet long, whose base is 200 feet long, and whose height is 80 feet, given that the dam is completely full with water.

Question 53

Find the hydrostatic force exerted on a dam in the shape of an isosceles triangle whose top is 250 feet wide and whose total height is 100 feet, given that the dam is completely full with water.

Question 54

Use definite integrals to find the centroid of the region between the graphs of $f ( x ) = 3 x$ and the x-axis on the interval [0, 3].

A) (2, 4)
B) (1.5, 3)
C) (2, 3)
D) (0, 3)

Question 55

Use definite integrals to find the centroid of the region bounded by the graphs of $f ( x ) = 2 - x ^ { 2 }$ , $f ( x ) = x$ , and the y-axis.

Question 56

Use definite integrals to find the centroid of the region between the graph of $f ( x ) = 2 \sqrt { x }$ and $g ( x ) = x$ on the interval [0, 4].

Question 57

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = \left( x ^ { 4 } + 2 \right) ^ { 2 }$$

Question 58

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = \frac { x - 2 } { \sqrt [ 3 ] { x } }$$

Question 59

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = \left( x ^ { 4 } + 2 \right) ^ { 2 } y$$

Question 60

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = x ^ { 4 } \cos ^ { 2 } 2 y$$

Question 61

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = x e ^ { x ^ { 2 } }$$

Question 62

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = x ^ { 3 } e ^ { - 2 y }$$

Question 63

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = \frac { 2 + 3 x } { x y }$$

Question 64

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = x ^ { 3 } y ^ { 2 }$$

Question 65

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = \frac { x y } { x ^ { 2 } + 1 }$$

Question 66

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = \frac { 3 } { x ^ { 2 } + 9 }$$

Question 67

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = \frac { 5 } { x y }$$

Question 68

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = y ( 1 - y )$$

Question 69

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = y ^ { 2 } \sin x$$

Question 70

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = e ^ { 2 x - y }$$

Question 71

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = x y + 2 x + 2 + y$ , $y ( 2 ) = - 1$

Question 72

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = y ^ { 3 } \cos x$ , $y \left( \frac { \pi } { 6 } \right) = 1$

Question 73

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = \frac { y } { x ^ { 2 } + 1 }$ , $y ( 1 ) = 1$

Question 74

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = \frac { x \sqrt { 1 - x ^ { 2 } } } { y }$ , $y ( 0 ) = 1$

Question 75

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = y ( 1 - y )$ , $y ( 0 ) = 2$

Question 76

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = 3 \sqrt { y }$ , $y ( 3 ) = 4$

Question 77

Use separation of variables to solve the initial value problem: $\frac { d T } { d t } = 20 - T$ , $T ( 0 ) = 15$

Question 78

Use separation of variables to solve the initial value problem: $\frac { d N } { d t } = 2 N$ , $N ( 1 ) = e ^ { 4 }$

Question 79

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }$ , $y ( 0 ) = 3$

Question 80

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = e ^ { 2 x - y }$ , $y ( 0 ) = 1$