Deck 32: More Applications of the Integral

Question 1

Find the length of the curve: $y^{2}=4 x$ from $y=0$ to $y=2$ . Use three significant digits.

Answer 1

2.30

Question 2

A bridge support arch has the equation $y=-\frac{1}{20}(x-30)^{2}+45$ . Find the length of the arch if the archis $60.0 \mathrm{~m}$ wide at the base.

Answer 2

$113 \mathrm{~m}$

Question 3

Find the length of the curve: $4 x+16 y^{2}=0$ from $y=0$ to $y=4$ . Use three significant digits.

Answer 3

64.3

Question 4

Find the length of the curve: $y=8 x^{2}+1$ from $x=0$ to 1 . Use three significant digits.

Question 5

Find the length of the curve: $x-y^{2}=12$ from $y=0$ to $y=3$ . Use three significant digits.

Question 6

Find the length of the curve: $x+2 y^{2}=1$ from $y=0$ to $y=1$ . Use three significant digits.

Question 7

Find the length of the curve $4 y^{2}=5 x$ from $x=1$ to $x=10$ . Use three significant digits.

Question 8

Find the length of the curve $2 y=x^{3}-1$ from $x=3$ to $x=5$ . Use three significant digits.

Question 9

Find the length of the curve $y^{2}=4 x^{3}$ from the origin to $(4,16)$ . Use three significant digits.

Question 10

Find the length of the curve $y=\sqrt{x^{3}}+1$ from $x=0$ to $x=9$ .

Question 11

A small walking bridge with a support arch $10.00 \mathrm{~m}$ wide and $2.00 \mathrm{~m}$ tall has the equation $y=-\frac{2}{25} x^{2}+2$ . Find the length of the $\operatorname{arch}(x=-5$ to $x=5)$ .

Question 12

Find the area of the surface generated by rotating the curve about the $x$ axis: $y^{2}=x+4$ from $x=0$ to $x=5$

Question 13

Find the area of the surface generated by revolving the curve around the $x$ axis: $y=4 x^{2}$ from $x=0$ to $x=2$ . Use three significant digits.

Question 14

Find the area of the surface generated by rotating the curve about the $y$ axis: $y^{2}=2 x-4$ for $y=1$ to $y=2$ . Use three significant digits.

Question 15

Find the area of the surface generated by rotating the curve about the $x$ axis: $y=x^{3}$ from $x=0$ to $x=1$ . Use three significant digits.

Question 16

Find the area of the surface generated by rotating the curve about the $x$ axis: $y=4-x^{2}$ from $x=0$ to $x=2$ . Use three significant digits.

Question 17

Find the area of the surface generated by rotating the curve $y=2 x^{2}+1$ from $x=0$ to $x=4$ about the $x$ -axis. Use three significant digits.

Question 18

Find the area of the surface generated by rotating the curve $y=2 \sqrt{x}$ from $x=\frac{1}{4}$ to $x=4$ about the $y$ -axis.

Question 19

Find the area of the surface generated by rotating the curve $y=\sqrt{x^{3}}$ from $x=0$ to $x=4$ about the $x$ -axis. Use three significant digits.

Question 20

Find the area of the surface generated by rotating the curve $x^{2}+y^{2}=4$ from $y=0$ to $y=1$ about the y-axis. Use three significant digits.

Question 21

Find the centroid of three particles of equal mass located at $(-2,6),(-4,-4)$ , and $(0,7)$ .

Question 22

Find the coordinates of the centroid of the area bounded by $y=\frac{1}{x}$ , $x$ axis, $x=1$ , and $x=e$ . Use three significant digits.

Question 23

Find the centroid of the area bounded by $y=x^{2}$ and $y=3 x$ .

Question 24

Find the distance from the origin to the centroid of the volume formed by rotating the area bounded by $y=9 x^{2}$ , the $y$ axis and $y=4$ around the $y$ axis. Use three significant digits.

Question 25

Find $\bar{x}$ and $\bar{y}$ of the area bounded by ${y=2 x^{2}}$ , the $x$ axis and $x=3$ .

Question 26

Find the centroid of three particles of equal mass located at $(1,-8),(-3,5)$ , and $(2,6)$ .

Question 27

Find the centroid of four particles of equal mass located at $(2,-3),(-5,7),(6,-1)$ , and $(-2,4)$ .

Question 28

Find $\bar{x}$ and $\bar{y}$ of the area bounded by $y=x^{2}$ and $y=x^{1 / 2}$ .

Question 29

Find the centroid of the area bounded by curves $y=x^{2}-4 x$ and $3 x+4 y-12=0$ .

Question 30

Find the centroid of the area bounded by the curves $3 y=x^{2}-9$ and $x^{2}+2 x+2 y-3=0$ .

Question 31

Find the centroid of the area bounded by $y=5 \sqrt{x}$ , x-axis, $x=1$ , and $x=4$ . Use three significant digits.

Question 32

Find the distance from the origin to the centroid of the volume formed by rotating the area bounded by $y=\sqrt{x}$ , the $x$ -axis and $x=9$ around the $x$ -axis.

Question 33

A rectangular plate, having a height of two metres and a width of five metres is submerged vertically such that its top edge is three metres below the surface of the water. Determine the force of the water acting on the plate to three significant digits. Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 34

A rectangular plate, having a height of $4.00 \mathrm{~m}$ and a width of $5.00 \mathrm{~m}$ is submerged vertically such that its top edge is $10.00 \mathrm{~m}$ below the surface of the water. Determine the pressure of the water acting on the plate to three significant digits. Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 35

A circular plate with a radius of three metres is submerged vertically half way in the water. Determine the force of the water acting on the plate to three significant digits. Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 36

A circular plate with a diameter of $1.00 \mathrm{~m}$ is submerged vertically in the water such that its centre is at a depth of $5.00 \mathrm{~m}$ . Determine the pressure of the water acting on the plate to three significant digits. Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 37

A circular plate having a radius of $2.00 \mathrm{~m}$ is submerged vertically such that its centre is $4.00 \mathrm{~m}$ below the surface of the water. Determine the force of the water acting on the plate to three significant digits. Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 38

A circular plate having a radius of $1.00 \mathrm{~m}$ is submerged vertically such that its center is $3.00 \mathrm{~m}$ below the surface of the water. Determine the pressure of the water acting on the plate to three significant digits. Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 39

A horizontal tank has ends in the shape of an ellipse $4.00 \mathrm{~m}$ wide and $2.00 \mathrm{~m}$ tall. Find the force on one end to three significant digits when the tank is half full of oil (density $=961 \mathrm{~kg} / \mathrm{m}^{3}$ ). Recall gravity is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 40

The electric force between two charges is equal to a constant divided by the distance between the charges squared. Two charges are separated by $1 \mathrm{~m}$ . The force between them is $10 \mathrm{~N}$ . How much work must be done to increase the distance between the charges to $3 \mathrm{~m}$ ?

Question 41

To hold a spring that has been stretched from its natural length of $20.0 \mathrm{~cm}$ to a length of $30.0 \mathrm{~cm}$ , a force of $20.0 \mathrm{~N}$ is required. Determine the work done when stretching the spring from $30.0 \mathrm{~cm}$ to $37.0 \mathrm{~cm}$ .

Question 42

Consider a particle $x$ metres from the origin, having a force of $x^{3}-x$ Newtons acting on it. Determine the work done to move the particle from $x=2$ to $x=5$ . Round your answer to two decimal places.

Question 43

An $8.00-\mathrm{cm}$ (free length) spring has a spring constant of $25.0 \mathrm{~N} / \mathrm{cm}$ . Find the work required to stretch it from $12.0 \mathrm{~cm}$ to $16.0 \mathrm{~cm}$ .

Question 44

A vertical cylindrical tank with a radius of $3.20 \mathrm{~m}$ and a height of $8.80 \mathrm{~m}$ is full of water. Find the work required to pump all the water to the top of the tank and out. Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is taken to be $9.81 \mathrm{~m} / \mathrm{s}^{2}$ . Use three significant digits.

Question 45

A vertical cylindrical tank that has a radius of $4.00 \mathrm{~m}$ and a height of $6.00 \mathrm{~m}$ . If the tank is three-quarters full of water, how much work is needed to pump the water to a height $1.00 \mathrm{~m}$ above the top of the tank? Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is taken to be $9.81 \mathrm{~m} / \mathrm{s}^{2}$ . Use three significant digits.

Question 46

A hemispherical tank with a radius of $3.50 \mathrm{~m}$ is full of water. Find the work needed to pump the water to a height $1.50 \mathrm{~m}$ above the top of the tank. Recall that the mass density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is taken to be $9.81 \mathrm{~m} / \mathrm{s}^{2}$ . Use three significant digits.

Question 47

To hold a spring that has been stretched from its natural length of $15.0 \mathrm{~cm}$ to a length of $20.0 \mathrm{~cm}$ , a force of $82.0 \mathrm{~N}$ is required. Determine the work done when stretching the spring from $20.0 \mathrm{~cm}$ to $25.0 \mathrm{~cm}$ .

Question 48

A vertical cylindrical tank with a radius of $2.50 \mathrm{~m}$ and a height of $6.25 \mathrm{~m}$ is full of oil. Find the work required to pump the oil to the top of the tank. Recall that the mass density of oil is $961 \mathrm{~kg} / \mathrm{m}^{3}$ and that gravity is taken to be $9.81 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 49

Find the moment of inertia of the area bounded by $x=0, y=9-3 x$ , and $y=0$ around the $y$ axis.

Question 50

Find the moment of inertia of the area bounded by $y^{3}=2 x^{3}+1, x=2$ , and the $y$ axis around the $x$ axis. Use three significant digits.

Question 51

Find the polar moment of inertia of the volume formed when the first quadrant area is rotated around the $x$ axis: bounded by $x+y=4$ . Use three significant digits.

Question 52

Find the moment of inertia about the $x$ and $y$ axes for the area under the curve $y=x^{2}$ , from $x=1$ to $x=2$ .

Question 53

Find the polar moment of inertia of the volume formed when the first-quadrant area with the following boundaries is rotated about the $x$ axis: $x+y=3$ , from $x=1$ to $x=2$ , and the $x$ axis. Use three significant digits.

Question 54

Find the moment of inertia of the area bounded by $x=1, x=3$ and $y=\frac{1}{x^{3}}$ , around the $y$ axis. Use three significant digits.

Question 55

Find the moment of inertia of the area bounded by $x=1, x=4$ and $2 y^{3}=x^{2}$ , around the $x$ axis.

Question 56

Find the polar moment of inertia of the volume formed when the area bounded by the curves $y=x^{2}$ and $y=4$ is rotated about the $x$ -axis.

Question 57

Find the polar moment of inertia of the volume formed when the first quadrant area is rotated around the $x$ axis: bounded by $y=x+2$ and $x=3$ . Use three significant digits.

Question 58

Find the moment of inertia of the area bounded by $y=3-2 x, x=0$ , and $y=0$ around the $y$ axis. Use three significant digits.