Question 1

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = 4x + 3, x = 2, x = 4, and y = 0 is revolved about the y-axis.

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Question 2

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by  , x = 0, y = 0, and x = 5 is revolved about the y-axis.

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Question 3

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y = x³ - 6x² + 8x over [0, 2] is revolved about the y-axis.

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Question 4

Use cylindrical shells to find the volume of the hemisphere that results when the region in the first quadrant enclosed by the circle x² + y² = 25 is revolved about the x-axis.

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Question 5

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = x², y = 4, and x = 0 is revolved about the x-axis.

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Question 6

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by x = 8y-y² and x = 0 is revolved about the x-axis.

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Question 7

Use cylindrical shells to find the volume of the solid generated when the region R under  over the interval [0, 7] is revolved about the x-axis.
A) 
B) 
C) 
D) 
E)

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Question 8

Use cylindrical shells to find the volume of the solid when the region enclosed by xy = 24, x + y = 10 is revolved about the y-axis.
A) 146.666 $\pi$
B) 0.01 $\pi$
C) 1.333 $\pi$
D) ( $\pi$)
E) 2.667 $\pi$

Accepted Answer

The volume of the solid can be found using the method of cylindrical shells. The region is bounded by the hyperbola xy = 24 and the line x + y = 10. Solving these equations simultaneously gives the points of intersection. The volume integral is set up with respect to x, and the limits of integration are determined by the x-values of the intersection points. The integral is evaluated to find the volume, which results in 2.667π.

Question 9

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x², x = 2, x = 4, y = 0 is revolved about the line x = 2. A) 45.333 $\pi$ B) 22.667 $\pi$ C) 330.667 $\pi$ D) 11.333 $\pi$ E) 2.011 $\pi$

Accepted Answer

The correct answer is A because the volume of the solid is found by integrating the area of the cross-sections of the solid with respect to the axis of revolution. The cross-sections are circles with radii equal to the distance from the axis of revolution to the curve y = x^2, which is 2 - x. The height of the cross-sections is the length of the curve y = x^2 between x = 2 and x = 4, which is 4 - 2 = 2. Therefore, the volume of the solid is:V = ∫[2,4] π(2 - x)^2 dx = π∫[2,4] (4 - 4x + x^2) dx = π[4x - 2x^2 + (1/3)x^3]_[2,4] = π(45.333)

Question 10

Use cylindrical shells to find the volume of the solid that results when the region enclosed by xy = 8, x + y = 9 is revolved about the y-axis.

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Question 11

A storage tank is designed by rotating y = -x² + 16, -4 $\le$ x $\le$ 4, about the x-axis where x and y are measured in meters. Use cylindrical shells to determine how many cubic meters the tank will hold. Approximate area to three decimal places.

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Question 12

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y² = 4x, x = 1, and y = 0 is revolved about the y-axis.

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Question 13

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

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Question 14

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = 2x + 3, x = 2, x = 5, and y = 0 is revolved about the y-axis.

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Question 15

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.

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Question 16

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

Accepted Answer

When a semicircle is revolved about the x-axis, it forms a sphere. Therefore, the region enclosed by revolving the semicircle 11ead0bc_8c97_f788_99a0_8fe68fa321d9_TB6988_11 about the x-axis is a sphere, which is represented by 11ead0bc_8c97_f789_99a0_e96519166208_TB6988_11.

Question 17

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y = x² and x = y² is revolved about the y-axis.

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Question 18

Use cylindrical shells to find the volume of the solid when the region enclosed by xy = 12, x + y = 8 is revolved about the x-axis.
A) 21.332 $\pi$
B) 0.01 $\pi$
C) 10.667 $\pi$
D) ( $\pi$)
E) 21.333 $\pi$

Accepted Answer

The volume of the solid can be found using the method of cylindrical shells. The region is bounded by the curves xy = 12 (a hyperbola) and x + y = 8 (a straight line). When revolved around the x-axis, the volume can be calculated by integrating the surface area of cylindrical shells. The correct calculation leads to the volume being 21.333π, which matches choice E.

Question 19

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y² = x⁷, x = 1, and y = 0 is revolved about the y-axis.

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Question 20

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

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

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = 4x + 3, x = 2, x = 4, and y = 0 is revolved about the y-axis.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by , x = 0, y = 0, and x = 5 is revolved about the y-axis.

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y = x3 - 6x2 + 8x over [0, 2] is revolved about the y-axis.

Use cylindrical shells to find the volume of the hemisphere that results when the region in the first quadrant enclosed by the circle x2 + y2 = 25 is revolved about the x-axis.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = x2, y = 4, and x = 0 is revolved about the x-axis.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by x = 8y-y2 and x = 0 is revolved about the x-axis.

Use cylindrical shells to find the volume of the solid generated when the region R under over the interval [0, 7] is revolved about the x-axis.

Use cylindrical shells to find the volume of the solid when the region enclosed by xy = 24, x + y = 10 is revolved about the y-axis.

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x2, x = 2, x = 4, y = 0 is revolved about the line x = 2.

Use cylindrical shells to find the volume of the solid that results when the region enclosed by xy = 8, x + y = 9 is revolved about the y-axis.

A storage tank is designed by rotating y = -x2 + 16, -4 ≤\le≤ x ≤\le≤ 4, about the x-axis where x and y are measured in meters. Use cylindrical shells to determine how many cubic meters the tank will hold. Approximate area to three decimal places.

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y2 = 4x, x = 1, and y = 0 is revolved about the y-axis.

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

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = 2x + 3, x = 2, x = 5, and y = 0 is revolved about the y-axis.

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.

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

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y = x2 and x = y2 is revolved about the y-axis.

Use cylindrical shells to find the volume of the solid when the region enclosed by xy = 12, x + y = 8 is revolved about the x-axis.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y2 = x7, x = 1, and y = 0 is revolved about the y-axis.

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

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y = x³ - 6x² + 8x over [0, 2] is revolved about the y-axis.

Use cylindrical shells to find the volume of the hemisphere that results when the region in the first quadrant enclosed by the circle x² + y² = 25 is revolved about the x-axis.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = x², y = 4, and x = 0 is revolved about the x-axis.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by x = 8y-y² and x = 0 is revolved about the x-axis.

Use cylindrical shells to find the volume of the solid when the region enclosed by y = x², x = 2, x = 4, y = 0 is revolved about the line x = 2.

A storage tank is designed by rotating y = -x² + 16, -4 $\le$ x $\le$ 4, about the x-axis where x and y are measured in meters. Use cylindrical shells to determine how many cubic meters the tank will hold. Approximate area to three decimal places.

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y² = 4x, x = 1, and y = 0 is revolved about the y-axis.

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

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.

Use cylindrical shells to find the volume of the solid that results when the region enclosed by y = x² and x = y² is revolved about the y-axis.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y² = x⁷, x = 1, and y = 0 is revolved about the y-axis.