Question 1

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = \sqrt [ 3 ] { x },$ the -axis, x = 1, and x = 5 about the x-axis?

Accepted Answer

The formula for the surface area of a solid of revolution about the x-axis is $2 \pi \int_a^b f(x) \sqrt{1 + (f'(x))^2} dx$, where $f(x)$ is the function being revolved and $f'(x)$ is its derivative. For $f(x) = x^{1/3}$, $f'(x) = \frac{1}{3}x^{-2/3}$. Substituting these into the formula gives the integral in choice C.

Question 2

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = x ^ { 4 } - 3 x,$ the x-axis, x = -2, and x = 0 about the x-axis?

Accepted Answer

The formula for the surface area of a solid of revolution about the x-axis is $2 \pi \int_a^b f(x) \sqrt{1 + (f'(x))^2} dx$, where $f(x)$ is the function being revolved and $f'(x)$ is its derivative. Here, $f(x) = x^4 - 3x$, and its derivative is $f'(x) = 4x^3 - 3$. Thus, the correct integral is $2 \pi \int_{-2}^{0} (x^4 - 3x) \sqrt{1 + (4x^3 - 3)^2} dx$.

Question 3

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 3 },$ the x-axis, x = -2, and x = 0 about the x-axis?

Accepted Answer

The correct integral for calculating the surface area of a solid formed by revolving a function $f(x)$ about the x-axis is given by $2 \pi \int y \sqrt{1 + (f'(x))^2} dx$, where $y = f(x)$. The derivative of $f(x) = (1 - x^2)^3$ is $f'(x) = -6x(1 - x^2)^2$, so the integral becomes $2 \pi \int (1 - x^2)^3 \sqrt{1 + 36x^2(1 - x^2)^4} dx$. The bounds should be from x = -2 to x = 0, but none of the options correctly reflect these bounds, indicating a potential error in the question or choices. However, option E most closely matches the formula for surface area despite the incorrect bounds.

Question 4

The work done by a variable force $F ( x ) = 40 - x$ Newtons that moves an object along a straight line in the direction of F from x = 5 meters to x = 20 meters, in Newton-meters, is

Accepted Answer

The answer of The work done by a variable force...

Question 5

The work done by a variable force $F ( x ) = x ^ { 2 } + 4$ Newtons that moves an object along a straight line in the direction of F from x = 1 meters to x = 3 meters, in Newton-meters, is

Accepted Answer

The work done by a variable force is given by the integral of the force over the distance. Here, $W = \int_{1}^{3} (x^2 + 4) dx$. Calculating this integral gives $W = \left[\frac{x^3}{3} + 4x\right]_{1}^{3} = \left(\frac{27}{3} + 12\right) - \left(\frac{1}{3} + 4\right) = 9 + 12 - \frac{1}{3} - 4 = \frac{27}{3} + 8 - \frac{1}{3} = \frac{26}{3} + 8 = \frac{50}{3}$ Newton-meters.

Question 6

The work done by a variable force $F ( x ) = ( 2 x + 1 ) ^ { 2 }$ Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

Accepted Answer

The answer of The work done by a variable force...

Question 7

The work done by a variable force $F ( x ) = ( 4 x - 1 ) ^ { 2 }$ Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

Accepted Answer

The work done by a variable force is calculated by integrating the force function over the distance. For $F(x) = (4x - 1)^2$, the work done from $x = 2$ to $x = 4$ is $\int_{2}^{4} (4x - 1)^2 dx$. Solving this integral gives $\frac{758}{3}$ Newton-meters.

Question 8

The work done by a winch winding in an 80-foot rope weighing 6 pounds per foot, in foot-pounds, is
​

Accepted Answer

The work done is calculated by the formula Work = Force x Distance. Here, the force is the weight of the rope, which is 6 pounds per foot for 80 feet, so the total weight (force) is 6 * 80 = 480 pounds. The distance is the length of the rope, which is 80 feet. Therefore, the work done is 480 pounds * 80 feet = 38,400 foot-pounds. However, since the rope's weight is distributed along its length and effectively halves as it is wound (since on average, half the rope is lifted at any point), the actual work done is half of 38,400, which is 19,200 foot-pounds.

Question 9

The work done by a winch winding in a 50-meter chain weighing 3 Newtons per meter, in Newton-meters, is
​

Accepted Answer

The work done is calculated by the formula Work = Force x Distance. Here, the force is the weight of the chain, which is 3 Newtons per meter for a 50-meter chain, so the total weight (force) is 3 N/m * 50 m = 150 N. The distance the force is applied over is 50 meters. Therefore, the work done is 150 N * 50 m = 7,500 Nm. However, since the chain is being wound up, it's effectively being lifted half the distance (as the average position of the chain moves from the ground to halfway up as it's wound), so the effective distance is 25 meters, not 50. Thus, the work done is 150 N * 25 m = 3,750 Nm.

Question 10

The work done by a winch winding in a 120-foot chain weighing 240 pounds, in foot-pounds, is
​

Accepted Answer

The work done is calculated by the formula Work = Force x Distance. Here, the force is the weight of the chain, which is 240 pounds, and the distance is the length of the chain, which is 120 feet. Therefore, Work = 240 pounds x 120 feet = 28,800 foot-pounds. However, since the chain is being wound in, the force is effectively halved over the distance (as the chain's center of mass moves closer to the winch), so the correct calculation is 28,800 foot-pounds / 2 = 14,400 foot-pounds.

Question 11

A spring in equilibrium is 0.8 meters long and has a spring constant of 5 Newtons/meter. The work done by the spring when it is compressed from equilibrium to a length of 0.5 meters, in Newton-meters, is
​

Accepted Answer

The answer of A spring in equilibrium is 0.8 meters...

Question 12

A spring in equilibrium is 0.8 meters long and an external force of 2 Newtons stretches the spring to a length of 1.2 meters. The work done by the spring force in stretching it from equilibrium to 3 meters, in Newton-meters, is
​

Accepted Answer

The answer of A spring in equilibrium is 0.8 meters...

Question 13

A spring in equilibrium is 10 inches long and an external force of 30 pounds stretches the spring to a length of 18 inches. The work done by the spring in stretching it from equilibrium to 12 inches, in inch-pounds, is
​

Accepted Answer

The work done by a spring can be calculated using the formula $ W = \frac{1}{2} k x^2 $, where $ k $ is the spring constant and $ x $ is the displacement from the equilibrium position. First, we need to find the spring constant $ k $. Given that a force of 30 pounds stretches the spring from 10 inches (its natural length) to 18 inches, the displacement $ x $ is 8 inches. Using Hooke's Law, $ F = kx $, we find $ k = \frac{F}{x} = \frac{30}{8} = 3.75 $ pounds per inch.To find the work done in stretching the spring from 10 inches to 12 inches, the displacement $ x $ is 2 inches. Using the work formula:$$ W = \frac{1}{2} k x^2 = \frac{1}{2} \times 3.75 \times 2^2 = \frac{1}{2} \times 3.75 \times 4 = 7.5 $$ inch-pounds.Since the work is done against the spring force, it is negative, making the correct answer -7.5 inch-pounds.

Question 14

A spring in equilibrium is 4 meters long and an external force of 8 Newtons stretches the spring to a length of 4.5 meters. The work done by the spring in stretching it from 4.5 meters to 5 meters, in Newton-meters, is
​

Accepted Answer

The spring constant k can be found using Hooke's Law: F = kx, where F = 8 N and x = 0.5 m (4.5 m - 4 m). So, k = 8 / 0.5 = 16 N/m. The work done by the spring is given by the formula W = 0.5 * k * (x2^2 - x1^2), where x1 = 0.5 m and x2 = 1 m (5 m - 4 m). Substituting the values, W = 0.5 * 16 * (1^2 - 0.5^2) = 0.5 * 16 * (1 - 0.25) = 0.5 * 16 * 0.75 = 6 Nm.

Question 15

A spring in equilibrium is 0.2 meters long and an external force of 0.5 Newtons stretches the spring to a length of 0.24 meters. The work done by the spring in stretching it from equilibrium to 0.28 meters, in Newton-meters, is
​

Accepted Answer

The answer of A spring in equilibrium is 0.2 meters...

Question 16

A spring in equilibrium is 12 centimeters long and an external force of 60 dynes compresses the spring to a length of 10 centimeters. The work done by the spring in compressing it from equilibrium to 8 centimeters, in dyne-centimeters, is
​

Accepted Answer

The work done by the spring is calculated using Hooke's Law and the formula for work done by a spring: W = 1/2 * k * (x1^2 - x2^2). First, find the spring constant k using F = k * x, where F = 60 dynes and x = 2 cm (12 cm - 10 cm), giving k = 30 dynes/cm. Then, calculate the work done from 12 cm to 8 cm: W = 1/2 * 30 * (4^2 - 0^2) = 240 dyne-cm.

Question 17

A right circular cylindrical water tank with height 30 meters and radius 5 meters is $\frac { 2 } { 3 }$ full. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in Newton-meters, is

Accepted Answer

The work required to pump water is calculated by the formula $W = \rho g h A d$, where $\rho$ is the density of water (1000 kg/m$^3$), $g$ is the acceleration due to gravity (9.8 m/s$^2$), $h$ is the height to which the water is lifted, $A$ is the cross-sectional area of the water ($\pi r^2$, with $r = 5$ meters), and $d$ is the depth of water being pumped ($\frac{2}{3} \times 30 = 20$ meters). The work done to pump the water to the top of the tank varies with the depth, so we integrate from 0 to 20 meters, considering the changing height $h$ from the water surface to the top of the tank. The total work is given by integrating the force (weight of water) times the distance it needs to be lifted. Simplifying, we find the total work to be approximately 98,000,000 Newton-meters, considering the integral accounts for the varying depth of water and the constant need to lift it over the top of the tank.

Question 18

A 25-meter long, 15-meter wide, and 5-meter deep rectangular water tank is full. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in Newton-meters, is

Accepted Answer

The work required to pump the water over the top of the tank can be calculated using the formula for work, $W = Fd$, where $F$ is the force applied and $d$ is the distance over which the force is applied. In this case, the force is equal to the weight of the water, which can be found by multiplying the volume of the water by the density of water and the acceleration due to gravity ($g = 9.8 m/s^2$). The volume of the tank is $25m \times 15m \times 5m = 1875 m^3$. The weight of the water is $1875 m^3 \times 1000 kg/m^3 \times 9.8 m/s^2 = 18,375,000 N$. The average distance the water needs to be lifted is half the depth of the tank, which is $2.5m$, since the water level drops as it is pumped out, but since it needs to be lifted over the top, the total distance is $5m + 2.5m = 7.5m$. Therefore, the work done is $18,375,000 N \times 7.5m = 137,812,500 N\cdot m$. However, none of the provided options match this calculation, indicating a mistake in my calculation regarding the average distance the water is lifted. The correct approach should consider the full depth since the question specifies pumping all the water over the top of the tank, making the distance $5m$, not the average distance. The correct calculation for work should directly consider the total volume of water, its weight due to gravity, and the height it needs to be lifted over the tank's top without averaging the lifting distance.Revising the calculation with the correct understanding: The total volume of water is $1875 m^3$, and the density of water is $1000 kg/m^3$, so the mass of the water is $1875 m^3 \times 1000 kg/m^3 = 1,875,000 kg$. The force due to gravity on this water is $1,875,000 kg \times 9.8 m/s^2 = 18,375,000 N$. The water needs to be lifted a distance of $5m$ (the depth of the tank) to be pumped over the top. Thus, the work done is $18,375,000 N \times 5m = 91,875,000 N\cdot m$. This calculation still does not match any of the provided options, indicating a fundamental misunderstanding in my explanation of how to calculate the work required correctly.The correct approach to find the work required to pump all the water over the top of the tank involves calculating the gravitational potential energy change for the entire volume of water being lifted to the tank's top. This involves the product of the mass of the water, gravitational acceleration, and the height it is lifted. The error in my explanation was in the misapplication of the height the water is lifted and misunderstanding the calculation process for the work required in this context. The correct calculation should directly apply the principles of energy work without incorrectly averaging the lifting distance for the water volume. Given the discrepancies in my calculations and explanations, the correct answer relies on understanding that the work done is equivalent to the gravitational potential energy change for lifting the water's entire volume to the tank's top, considering the correct height and the total mass of the water.

Question 19

A hemispherical water tank of radius 6 meters is filled with water to a depth of 4 meters. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in kg-meters, is

Accepted Answer

The answer of A hemispherical water tank of radius 6...

Question 20

A right circular cylindrical storage tank with height 10 meters and radius 8 meters is full of gasoline. Assume the density of gasoline is 720 kg/cu m. The work required to pump all the gasoline over the top of the tank, in kg-meters, is

Accepted Answer

The work required to pump the gasoline over the top of the tank can be calculated using the formula for work, $W = \rho g h A d$, where $\rho$ is the density of the fluid (720 kg/m³ for gasoline), $g$ is the acceleration due to gravity (9.8 m/s², but it cancels out since we're asked for work in kg-meters), $h$ is the height at which the fluid is being lifted (which varies from 0 at the bottom to 10 meters at the top), $A$ is the cross-sectional area of the cylinder ($\pi r^2 = \pi \times 8^2 = 64\pi$ m²), and $d$ is the depth of the fluid element being considered.The work done to lift a thin slice of gasoline at a height $y$ to the top of the tank is $dW = \rho A g (10 - y) dy$, integrating this from $y = 0$ to $y = 10$ gives the total work:$$W = \int_0^{10} 720 \times 64\pi \times (10 - y) dy = 720 \times 64\pi \times \left[10y - \frac{y^2}{2}\right]_0^{10}$$$$W = 720 \times 64\pi \times \left[100 - \frac{100}{2}\right] = 720 \times 64\pi \times 50$$$$W = 720 \times 3200\pi = 2,304,000\pi$$However, this calculation does not match any of the provided options, indicating a mistake in the calculation process. The correct approach involves calculating the volume of gasoline and then multiplying by the average height it needs to be lifted (which is 5 meters, since the work required is equivalent to lifting the entire volume to the midpoint of the tank, then over the edge):Volume of the cylinder $V = \pi r^2 h = \pi \times 8^2 \times 10 = 640\pi$ m³.The mass of the gasoline is $m = \rho V = 720 \times 640\pi = 460,800\pi$ kg.The work required to lift this mass over the top of the tank (5 meters on average) is $W = mgh = 460,800\pi \times 5 = 2,304,000\pi$ kg-meters.This calculation still does not directly lead to the correct answer, indicating a misunderstanding in the explanation process. The correct method to find the work involves integrating the force needed to lift each slice of gasoline over the height of the tank, considering the density and the gravitational force. The discrepancy in the explanation suggests a need to correctly apply the principles of calculus and physics to find the total work done, which involves integrating the weight of each infinitesimal volume of gasoline over its lifting distance. Given the options and the typical approach to such a problem, the correct answer involves calculating the total energy required to move the entire volume of gasoline, taking into account the density and the height of the tank. The mistake in the initial explanation points to a miscalculation or misunderstanding of the problem's requirements. The correct calculation should directly relate to the volume of the tank, the density of gasoline, and the height over which the gasoline must be lifted, leading to one of the provided choices.

Quiz 7: Applications of the Integral

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by f(x) =x3,f ( x ) = \sqrt [ 3 ] { x },f(x) =3x​, the -axis, x = 1, and x = 5 about the x-axis?

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by f(x) =x4−3x,f ( x ) = x ^ { 4 } - 3 x,f(x) =x4−3x, the x-axis, x = -2, and x = 0 about the x-axis?

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by f(x) =(1−x2) 3,f ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 3 },f(x) =(1−x2) 3, the x-axis, x = -2, and x = 0 about the x-axis?

The work done by a variable force F(x) =40−xF ( x ) = 40 - xF(x) =40−x Newtons that moves an object along a straight line in the direction of F from x = 5 meters to x = 20 meters, in Newton-meters, is

The work done by a variable force F(x) =x2+4F ( x ) = x ^ { 2 } + 4F(x) =x2+4 Newtons that moves an object along a straight line in the direction of F from x = 1 meters to x = 3 meters, in Newton-meters, is

The work done by a variable force F(x) =(2x+1) 2F ( x ) = ( 2 x + 1 ) ^ { 2 }F(x) =(2x+1) 2 Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

The work done by a variable force F(x) =(4x−1) 2F ( x ) = ( 4 x - 1 ) ^ { 2 }F(x) =(4x−1) 2 Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

The work done by a winch winding in an 80-foot rope weighing 6 pounds per foot, in foot-pounds, is ​

The work done by a winch winding in a 50-meter chain weighing 3 Newtons per meter, in Newton-meters, is ​

The work done by a winch winding in a 120-foot chain weighing 240 pounds, in foot-pounds, is ​

A spring in equilibrium is 0.8 meters long and has a spring constant of 5 Newtons/meter. The work done by the spring when it is compressed from equilibrium to a length of 0.5 meters, in Newton-meters, is ​

A spring in equilibrium is 0.8 meters long and an external force of 2 Newtons stretches the spring to a length of 1.2 meters. The work done by the spring force in stretching it from equilibrium to 3 meters, in Newton-meters, is ​

A spring in equilibrium is 10 inches long and an external force of 30 pounds stretches the spring to a length of 18 inches. The work done by the spring in stretching it from equilibrium to 12 inches, in inch-pounds, is ​

A spring in equilibrium is 4 meters long and an external force of 8 Newtons stretches the spring to a length of 4.5 meters. The work done by the spring in stretching it from 4.5 meters to 5 meters, in Newton-meters, is ​

A spring in equilibrium is 0.2 meters long and an external force of 0.5 Newtons stretches the spring to a length of 0.24 meters. The work done by the spring in stretching it from equilibrium to 0.28 meters, in Newton-meters, is ​

A spring in equilibrium is 12 centimeters long and an external force of 60 dynes compresses the spring to a length of 10 centimeters. The work done by the spring in compressing it from equilibrium to 8 centimeters, in dyne-centimeters, is ​

A right circular cylindrical water tank with height 30 meters and radius 5 meters is 23\frac { 2 } { 3 }32​ full. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in Newton-meters, is

A 25-meter long, 15-meter wide, and 5-meter deep rectangular water tank is full. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in Newton-meters, is

A hemispherical water tank of radius 6 meters is filled with water to a depth of 4 meters. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in kg-meters, is

A right circular cylindrical storage tank with height 10 meters and radius 8 meters is full of gasoline. Assume the density of gasoline is 720 kg/cu m. The work required to pump all the gasoline over the top of the tank, in kg-meters, is

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = \sqrt [ 3 ] { x },$ the -axis, x = 1, and x = 5 about the x-axis?

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = x ^ { 4 } - 3 x,$ the x-axis, x = -2, and x = 0 about the x-axis?

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 3 },$ the x-axis, x = -2, and x = 0 about the x-axis?

The work done by a variable force $F ( x ) = 40 - x$ Newtons that moves an object along a straight line in the direction of F from x = 5 meters to x = 20 meters, in Newton-meters, is

The work done by a variable force $F ( x ) = x ^ { 2 } + 4$ Newtons that moves an object along a straight line in the direction of F from x = 1 meters to x = 3 meters, in Newton-meters, is

The work done by a variable force $F ( x ) = ( 2 x + 1 ) ^ { 2 }$ Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

The work done by a variable force $F ( x ) = ( 4 x - 1 ) ^ { 2 }$ Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

The work done by a winch winding in an 80-foot rope weighing 6 pounds per foot, in foot-pounds, is

The work done by a winch winding in a 50-meter chain weighing 3 Newtons per meter, in Newton-meters, is

The work done by a winch winding in a 120-foot chain weighing 240 pounds, in foot-pounds, is

A spring in equilibrium is 0.8 meters long and has a spring constant of 5 Newtons/meter. The work done by the spring when it is compressed from equilibrium to a length of 0.5 meters, in Newton-meters, is

A spring in equilibrium is 0.8 meters long and an external force of 2 Newtons stretches the spring to a length of 1.2 meters. The work done by the spring force in stretching it from equilibrium to 3 meters, in Newton-meters, is

A spring in equilibrium is 10 inches long and an external force of 30 pounds stretches the spring to a length of 18 inches. The work done by the spring in stretching it from equilibrium to 12 inches, in inch-pounds, is

A spring in equilibrium is 4 meters long and an external force of 8 Newtons stretches the spring to a length of 4.5 meters. The work done by the spring in stretching it from 4.5 meters to 5 meters, in Newton-meters, is

A spring in equilibrium is 0.2 meters long and an external force of 0.5 Newtons stretches the spring to a length of 0.24 meters. The work done by the spring in stretching it from equilibrium to 0.28 meters, in Newton-meters, is

A spring in equilibrium is 12 centimeters long and an external force of 60 dynes compresses the spring to a length of 10 centimeters. The work done by the spring in compressing it from equilibrium to 8 centimeters, in dyne-centimeters, is

A right circular cylindrical water tank with height 30 meters and radius 5 meters is $\frac { 2 } { 3 }$ full. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in Newton-meters, is