Question 1

Find the arc length of the curve  from x = 0 to x = 3.

Accepted Answer

A)  17.436 
B)  6.495 
C)  19.948 
D)  7.783 
E)  18.739

Question 2

The face of the dam shown in the figure is an inclined rectangle. If the dam is 60 feet wide, the face is 30 feet in length and $\theta$= 30°, find the fluid force on the face when the water (weight density 62.4 lbs/ft³) is level with the top of the dam.

Accepted Answer

The answer of The face of the dam shown in...

Question 3

Find the area of the surface generated when  from x = 1 to x = 3 is rotated around the x-axis.

Accepted Answer

The answer of Find the area of the surface generated...

Question 4

Find the arc length of the curve  from x = 1 to x = 2.

Accepted Answer

The answer of Find the arc length of the curve...

Question 5

Use cylindrical shells to find the volume of the solid that results when the area of the smaller region enclosed by y² = 9x, y = 3, and x = 4 is revolved about the y-axis

Accepted Answer

The answer of Use cylindrical shells to find the volume...

Question 6

A right triangle is submerged vertically with one side parallel to the surface in a liquid of density$\rho$. The triangle has a leg that is 16 m long located at the surface and a leg 16 m long straight down. Find the force exerted on the triangular surface, in terms of the density. Neglect the effect of the atmosphere above the liquid.

Accepted Answer

The answer of A right triangle is submerged vertically with...

Question 7

A conical tank has a diameter of 12 feet and is 30 feet deep. If the tank is filled with water of density 62.4lbs/ft³, how much work is required to pump the water over the top?

Accepted Answer

The answer of A conical tank has a diameter of...

Question 8

A force of  lb is applied to an object as it moves from x = -1 ft to x = 2 ft. How much work is done?

Accepted Answer

The answer of A force of  lb is applied...

Question 9

Find the area of the region enclosed by  , the x axis, x = 3, and x = 6.

Accepted Answer

The answer of Find the area of the region enclosed...

Question 10

Find a vertical line x = k that divides the area enclosed by  , y = 0 and x = 5 into equal parts. Approximate area to three decimal places.

Accepted Answer

A)  2.5 
B)  3.150 
C)  10 
D)  3.536 
E)  7.937

Question 11

The area of the surface generated by revolving the curve x = 3 cos t, y = 3 sin t, 0 $\le$  t $\le$ $\pi$ about the x-axis is

Accepted Answer

A)  9 $\pi$ 
B)  36 
C)  18 $\pi$ 
D)  45 $\pi$ 
E)  36 $\pi$

Question 12

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y = 16 -x² and y = x + 4 is revolved about the x axis.

Accepted Answer

The answer of Use the method of washers to find...

Question 13

Find the value of the solid that is obtained when the region under the curve y = x2 + 2x over the interval [1, 2] is revolved about the x-axis. Approximate area to three decimal places.

Accepted Answer

A)  33.067 
B)  2.533 
C)  30.533 
D)  95.923 
E)  103.882

Question 14

Answer true or false. A 12 in diameter glass circular window on the side of a submarine has the same force acting on the top half as on the bottom half.

Accepted Answer

A) True 
 B)False

Question 15

A flat rectangular plate is submerged horizontally in water to a depth of 3.2 ft. If the top surface of the plate has an area of 50 ft², and the liquid in which it is submerged is water, find the force on the top of the plate. Neglect the effect of the atmosphere above the liquid. (The density of water is 62.6 lb/ft³.)

Accepted Answer

The answer of A flat rectangular plate is submerged horizontally...

Question 16

Answer true or false. The region enclosed by revolving the semicircle  about the x-axis is  .

Accepted Answer

A) True 
 B)False

Question 17

If the gravitational force is proportional to x-8, the work it does is proportional to

Accepted Answer

A)  x-9 
B)  x-7 
C)  x-8 
D)  x7 
E)  x9

Question 18

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x2, x = 1, x = 2, y = 0 is revolved about the y-axis.

Accepted Answer

A)  15 $\pi$ 
B)   
C)   
D)   
E)

Question 19

Find the work done when a variable force of F(x) = 26x N in the positive x direction moves an object from x = 0 m to x = 3 m.

Accepted Answer

A)  78 J 
B)  39 J 
C)  234 J 
D)  117 J 
E)  4.5 J

Question 20

Answer true or false. The amount of work needed to lift a liquid of density 1.38 kg/m3 from a spherical tank of radius 7 m is  .

Accepted Answer

A) True 
 B)False

Exam 6: Applications of the Definite Integral in Geometry, Science and Engineering

Find the arc length of the curve from x = 0 to x = 3.

Find the area of the surface generated when from x = 1 to x = 3 is rotated around the x-axis.

Find the arc length of the curve from x = 1 to x = 2.

Use cylindrical shells to find the volume of the solid that results when the area of the smaller region enclosed by y² = 9x, y = 3, and x = 4 is revolved about the y-axis

A conical tank has a diameter of 12 feet and is 30 feet deep. If the tank is filled with water of density 62.4lbs/ft³, how much work is required to pump the water over the top?

A force of lb is applied to an object as it moves from x = -1 ft to x = 2 ft. How much work is done?

Find the area of the region enclosed by , the x axis, x = 3, and x = 6.

Find a vertical line x = k that divides the area enclosed by , y = 0 and x = 5 into equal parts. Approximate area to three decimal places.

The area of the surface generated by revolving the curve x = 3 cos t, y = 3 sin t, 0 $\le$ t $\le$ $\pi$ about the x-axis is

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y = 16 -x² and y = x + 4 is revolved about the x axis.

Find the value of the solid that is obtained when the region under the curve y = x² + 2x over the interval [1, 2] is revolved about the x-axis. Approximate area to three decimal places.

Answer true or false. A 12 in diameter glass circular window on the side of a submarine has the same force acting on the top half as on the bottom half.

Answer true or false. The region enclosed by revolving the semicircle about the x-axis is .

If the gravitational force is proportional to x^-⁸, the work it does is proportional to

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x², x = 1, x = 2, y = 0 is revolved about the y-axis.

Find the work done when a variable force of F(x) = 26x N in the positive x direction moves an object from x = 0 m to x = 3 m.

Answer true or false. The amount of work needed to lift a liquid of density 1.38 kg/m³ from a spherical tank of radius 7 m is .

Exam 6: Applications of the Definite Integral in Geometry, Science and Engineering

Find the arc length of the curve from x = 0 to x = 3.

The face of the dam shown in the figure is an inclined rectangle. If the dam is 60 feet wide, the face is 30 feet in length and θ\thetaθ = 30°, find the fluid force on the face when the water (weight density 62.4 lbs/ft3) is level with the top of the dam.

Find the area of the surface generated when from x = 1 to x = 3 is rotated around the x-axis.

Find the arc length of the curve from x = 1 to x = 2.

Use cylindrical shells to find the volume of the solid that results when the area of the smaller region enclosed by y2 = 9x, y = 3, and x = 4 is revolved about the y-axis

A conical tank has a diameter of 12 feet and is 30 feet deep. If the tank is filled with water of density 62.4lbs/ft3, how much work is required to pump the water over the top?

A force of lb is applied to an object as it moves from x = -1 ft to x = 2 ft. How much work is done?

Find the area of the region enclosed by , the x axis, x = 3, and x = 6.

Find a vertical line x = k that divides the area enclosed by , y = 0 and x = 5 into equal parts. Approximate area to three decimal places.

The area of the surface generated by revolving the curve x = 3 cos t, y = 3 sin t, 0 ≤\le≤ t ≤\le≤ π\piπ about the x-axis is

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y = 16 -x2 and y = x + 4 is revolved about the x axis.

Find the value of the solid that is obtained when the region under the curve y = x2 + 2x over the interval [1, 2] is revolved about the x-axis. Approximate area to three decimal places.

Answer true or false. A 12 in diameter glass circular window on the side of a submarine has the same force acting on the top half as on the bottom half.

Answer true or false. The region enclosed by revolving the semicircle about the x-axis is .

If the gravitational force is proportional to x-8, the work it does is proportional to

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x2, x = 1, x = 2, y = 0 is revolved about the y-axis.

Find the work done when a variable force of F(x) = 26x N in the positive x direction moves an object from x = 0 m to x = 3 m.

Answer true or false. The amount of work needed to lift a liquid of density 1.38 kg/m3 from a spherical tank of radius 7 m is .

Use cylindrical shells to find the volume of the solid that results when the area of the smaller region enclosed by y² = 9x, y = 3, and x = 4 is revolved about the y-axis

A conical tank has a diameter of 12 feet and is 30 feet deep. If the tank is filled with water of density 62.4lbs/ft³, how much work is required to pump the water over the top?

The area of the surface generated by revolving the curve x = 3 cos t, y = 3 sin t, 0 $\le$ t $\le$ $\pi$ about the x-axis is

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y = 16 -x² and y = x + 4 is revolved about the x axis.

Find the value of the solid that is obtained when the region under the curve y = x² + 2x over the interval [1, 2] is revolved about the x-axis. Approximate area to three decimal places.

If the gravitational force is proportional to x^-⁸, the work it does is proportional to

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x², x = 1, x = 2, y = 0 is revolved about the y-axis.

Answer true or false. The amount of work needed to lift a liquid of density 1.38 kg/m³ from a spherical tank of radius 7 m is .