Question 1

Use a graphing utility to find the area of the region enclosed by the curves x = sin 4 y, x = 0, y = 0, y = - $\pi$/4.

Accepted Answer

The answer of Use a graphing utility to find the...

Question 2

Find the area of the region enclosed by x = y² - 5y and x = y.

Accepted Answer

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

Question 3

A spring whose natural length is 18 inches exerts a force of 7.2 pounds when stretched to 26 inches. How much work is required to stretch the spring 16 inches beyond its natural length?

Accepted Answer

The answer of A spring whose natural length is 18...

Question 4

Find the value of the solid that is obtained when the region enclosed by the curves x = y3 , x = 0, y = 1, y = 3 is revolved about the y-axis.

Accepted Answer

A)   
B)  2,186 $\pi$ 
C)   
D)  7 $\pi$ 
E)

Question 5

Find the centroid of the region enclosed by y = -x² + 8 and from x = 0 to x = 1.

Accepted Answer

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

Question 6

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves y = cos6x, x = 0,  about the x-axis. Approximate area to three decimal places.

Accepted Answer

A)  2.899 
B)  0.936 
C)  0.362 
D)  0.250 
E)  1.007

Question 7

Find the area of the surface that is generated by revolving the portion of the curve y = x² - 2 between x = 0 and x = 1 about the x-axis. Approximate area to three decimal places.

Accepted Answer

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

Question 8

A horizontal cylindrical tank of diameter 12 feet is half full of a chemical (weight density 55 lbs/ft³). Calculate the force against one end.

Accepted Answer

The answer of A horizontal cylindrical tank of diameter 12...

Question 9

Let a hemisphere of radius 5 be cut by a plane parallel to the base of the hemisphere thus forming a segment of height 1 (in the top portion of the hemisphere). Find its volume using cylindrical shells.

Accepted Answer

The answer of Let a hemisphere of radius 5 be...

Question 10

Assume 10 J of work stretch a spring 2 cm. Find the spring constant in J/cm.

Accepted Answer

A)  0.2 
B)  5 
C)  20 
D)  0.05 
E)  6.25

Question 11

The steeple of a church is constructed in the form of a pyramid 45 feet high. The cross sections are all squares and the base is a square of side 15 feet. Find the volume of the steeple.

Accepted Answer

The answer of The steeple of a church is constructed...

Question 12

Use cylindrical shells to find the volume of the cone generated when the triangle with vertices (0, 0), (3, 0), (0, 12), where r > 0 and h > 0 is revolved about the y-axis.

Accepted Answer

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

Question 13

Find the area of the region enclosed by y = 4x -x² and y = 3.

Accepted Answer

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

Question 14

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x - 8, y = -x, x = 8, y = 0 is revolved about the y-axis.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 15

A flat triangular plate whose dimensions are 13, 13, and 10 feet is submerged in water (weight density 62.4 lbs/ft³) so that its longer side is at the surface and parallel to it. Find the force against the surface of the plate.

Accepted Answer

The answer of A flat triangular plate whose dimensions are...

Question 16

Answer true or false. The volume of the solid that results when the region enclosed by the curves y = x2 and x = y is revolved about x = 1, correct to three decimals, is 0.524.

Accepted Answer

A) True 
 B)False

Question 17

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y² = x³, x = 1, and y = 0 is revolved about the x-axis.

Accepted Answer

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

Question 18

Find the area of the region enclosed by y = x² - 4x and y = 16 - x².

Accepted Answer

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

Question 19

The base of a solid is a circle of radius 5. All sections that are perpendicular to the diameter are squares. Find the volume of the solid.

Accepted Answer

The answer of The base of a solid is a...

Question 20

Find the arc length of the curve  from y = 1 to y = 3.

Accepted Answer

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

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

Use a graphing utility to find the area of the region enclosed by the curves x = sin 4 y, x = 0, y = 0, y = - $\pi$ /4.

Find the area of the region enclosed by x = y² - 5y and x = y.

A spring whose natural length is 18 inches exerts a force of 7.2 pounds when stretched to 26 inches. How much work is required to stretch the spring 16 inches beyond its natural length?

Find the value of the solid that is obtained when the region enclosed by the curves x = y³ , x = 0, y = 1, y = 3 is revolved about the y-axis.

Find the centroid of the region enclosed by y = -x² + 8 and from x = 0 to x = 1.

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves y = cos⁶x, x = 0, about the x-axis. Approximate area to three decimal places.

Find the area of the surface that is generated by revolving the portion of the curve y = x² - 2 between x = 0 and x = 1 about the x-axis. Approximate area to three decimal places.

A horizontal cylindrical tank of diameter 12 feet is half full of a chemical (weight density 55 lbs/ft³). Calculate the force against one end.

Let a hemisphere of radius 5 be cut by a plane parallel to the base of the hemisphere thus forming a segment of height 1 (in the top portion of the hemisphere). Find its volume using cylindrical shells.

Assume 10 J of work stretch a spring 2 cm. Find the spring constant in J/cm.

The steeple of a church is constructed in the form of a pyramid 45 feet high. The cross sections are all squares and the base is a square of side 15 feet. Find the volume of the steeple.

Use cylindrical shells to find the volume of the cone generated when the triangle with vertices (0, 0), (3, 0), (0, 12), where r > 0 and h > 0 is revolved about the y-axis.

Find the area of the region enclosed by y = 4x -x² and y = 3.

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x - 8, y = -x, x = 8, y = 0 is revolved about the y-axis.

A flat triangular plate whose dimensions are 13, 13, and 10 feet is submerged in water (weight density 62.4 lbs/ft³) so that its longer side is at the surface and parallel to it. Find the force against the surface of the plate.

Answer true or false. The volume of the solid that results when the region enclosed by the curves y = x² and x = y is revolved about x = 1, correct to three decimals, is 0.524.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y² = x³, x = 1, and y = 0 is revolved about the x-axis.

Find the area of the region enclosed by y = x² - 4x and y = 16 - x².

The base of a solid is a circle of radius 5. All sections that are perpendicular to the diameter are squares. Find the volume of the solid.

Find the arc length of the curve from y = 1 to y = 3.

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

Use a graphing utility to find the area of the region enclosed by the curves x = sin 4 y, x = 0, y = 0, y = - π\piπ /4.

Find the area of the region enclosed by x = y2 - 5y and x = y.

A spring whose natural length is 18 inches exerts a force of 7.2 pounds when stretched to 26 inches. How much work is required to stretch the spring 16 inches beyond its natural length?

Find the value of the solid that is obtained when the region enclosed by the curves x = y3 , x = 0, y = 1, y = 3 is revolved about the y-axis.

Find the centroid of the region enclosed by y = -x2 + 8 and from x = 0 to x = 1.

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves y = cos6x, x = 0, about the x-axis. Approximate area to three decimal places.

Find the area of the surface that is generated by revolving the portion of the curve y = x2 - 2 between x = 0 and x = 1 about the x-axis. Approximate area to three decimal places.

A horizontal cylindrical tank of diameter 12 feet is half full of a chemical (weight density 55 lbs/ft3). Calculate the force against one end.

Let a hemisphere of radius 5 be cut by a plane parallel to the base of the hemisphere thus forming a segment of height 1 (in the top portion of the hemisphere). Find its volume using cylindrical shells.

Assume 10 J of work stretch a spring 2 cm. Find the spring constant in J/cm.

The steeple of a church is constructed in the form of a pyramid 45 feet high. The cross sections are all squares and the base is a square of side 15 feet. Find the volume of the steeple.

Use cylindrical shells to find the volume of the cone generated when the triangle with vertices (0, 0), (3, 0), (0, 12), where r > 0 and h > 0 is revolved about the y-axis.

Find the area of the region enclosed by y = 4x -x2 and y = 3.

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x - 8, y = -x, x = 8, y = 0 is revolved about the y-axis.

A flat triangular plate whose dimensions are 13, 13, and 10 feet is submerged in water (weight density 62.4 lbs/ft3) so that its longer side is at the surface and parallel to it. Find the force against the surface of the plate.

Answer true or false. The volume of the solid that results when the region enclosed by the curves y = x2 and x = y is revolved about x = 1, correct to three decimals, is 0.524.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y2 = x3, x = 1, and y = 0 is revolved about the x-axis.

Find the area of the region enclosed by y = x2 - 4x and y = 16 - x2.

The base of a solid is a circle of radius 5. All sections that are perpendicular to the diameter are squares. Find the volume of the solid.

Find the arc length of the curve from y = 1 to y = 3.

Use a graphing utility to find the area of the region enclosed by the curves x = sin 4 y, x = 0, y = 0, y = - $\pi$ /4.

Find the area of the region enclosed by x = y² - 5y and x = y.

Find the value of the solid that is obtained when the region enclosed by the curves x = y³ , x = 0, y = 1, y = 3 is revolved about the y-axis.

Find the centroid of the region enclosed by y = -x² + 8 and from x = 0 to x = 1.

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves y = cos⁶x, x = 0, about the x-axis. Approximate area to three decimal places.

Find the area of the surface that is generated by revolving the portion of the curve y = x² - 2 between x = 0 and x = 1 about the x-axis. Approximate area to three decimal places.

A horizontal cylindrical tank of diameter 12 feet is half full of a chemical (weight density 55 lbs/ft³). Calculate the force against one end.

Find the area of the region enclosed by y = 4x -x² and y = 3.

A flat triangular plate whose dimensions are 13, 13, and 10 feet is submerged in water (weight density 62.4 lbs/ft³) so that its longer side is at the surface and parallel to it. Find the force against the surface of the plate.

Answer true or false. The volume of the solid that results when the region enclosed by the curves y = x² and x = y is revolved about x = 1, correct to three decimals, is 0.524.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y² = x³, x = 1, and y = 0 is revolved about the x-axis.

Find the area of the region enclosed by y = x² - 4x and y = 16 - x².