Question 1

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x², y = 0, and x = 3 is revolved about x = 3.

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Question 2

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x⁵, x = 1, and y = 0 is revolved about the line x = 1.

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Question 3

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y² = x³, x = 4, and y = 0 is revolved about the y axis.

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Question 4

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = (7 - x)², x = 7, and y = 1 is revolved about the line x = 7.

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Question 5

Use the method of washers to find the volume of the solid that results when the area enclosed by y² = 16x, y = 5, and x = 7 is revolved about the x axis.

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Question 6

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x³, x = 1, and y = -8 is revolved about the line y = -1.

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Question 7

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x⁴, x = 0, and y = 3 is revolved about the y axis.

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Question 8

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.

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Question 9

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by  , y = 0, and x = 16 is revolved about the y axis.

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Question 10

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y² = x³, x = 3, and y = 0 is revolved about the x axis.

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Question 11

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.

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Question 12

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.

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Question 13

Find the volume of the solid that results when the area of the region enclosed by x + y = 4, y = 0, x = 0 is revolved about the y axis

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Question 14

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y² = 9x, x = 16, and y = 0 is revolved about the y axis.

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Question 15

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.

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The given curves are y = x^2 and x = y, which intersect at points (0,0) and (1,1). When the region enclosed by these curves is revolved about x = 1, the volume of the resulting solid can be calculated using the method of cylindrical shells or the disk method. However, the exact volume calculation would depend on the method used and the integration setup, and without performing the specific calculations, the provided volume of 0.524.. does not directly correspond to a common result for this setup. The actual calculation involves integrating the area of the rotated region, which typically does not yield a simple value like 0.524 without more context or a specific calculation.

Question 16

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y = x², y = 1, and x = 0 is revolved about the x axis.

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Question 17

Answer true or false: A right-circular cylinder of radius 8cm contains a hollow sphere of radius 7 cm. If the cylinder is filled to a height h with water and the sphere floats so that its highest point is 4cm above the water level, there is cm³ of water in the cylinder.

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Question 18

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves x = y², x = y + 20, about the y-axis. Approximate area to three decimal places.

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Question 19

Answer true or false. The volume of the solid that results when the region enclosed by the curves x = y², x = y⁵ is revolved about the y-axis is given by .

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Question 20

Find the volume of the hemisphere created by rotating the area enclosed by the circle x² + y² = 4 and above y = 0, around the y-axis. Approximate area to three decimal places.

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

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x2, y = 0, and x = 3 is revolved about x = 3.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x5, x = 1, and y = 0 is revolved about the line x = 1.

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y2 = x3, x = 4, and y = 0 is revolved about the y axis.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = (7 - x)2 , x = 7, and y = 1 is revolved about the line x = 7.

Use the method of washers to find the volume of the solid that results when the area enclosed by y2 = 16x, y = 5, and x = 7 is revolved about the x axis.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x3, x = 1, and y = -8 is revolved about the line y = -1.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x4, x = 0, and y = 3 is revolved about the y axis.

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.

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by , y = 0, and x = 16 is revolved about the y axis.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y2 = x3, x = 3, and y = 0 is revolved about the x axis.

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.

Find the volume of the solid that results when the area of the region enclosed by x + y = 4, y = 0, x = 0 is revolved about the y axis

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y2 = 9x, x = 16, and y = 0 is revolved about the y axis.

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 the method of washers to find the volume of the solid that results when the area of the region enclosed by y = x2, y = 1, and x = 0 is revolved about the x axis.

Answer true or false: A right-circular cylinder of radius 8cm contains a hollow sphere of radius 7 cm. If the cylinder is filled to a height h with water and the sphere floats so that its highest point is 4cm above the water level, there is cm3 of water in the cylinder.

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves x = y2, x = y + 20, about the y-axis. Approximate area to three decimal places.

Answer true or false. The volume of the solid that results when the region enclosed by the curves x = y2, x = y5 is revolved about the y-axis is given by .

Find the volume of the hemisphere created by rotating the area enclosed by the circle x2 + y2 = 4 and above y = 0, around the y-axis. Approximate area to three decimal places.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x², y = 0, and x = 3 is revolved about x = 3.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x⁵, x = 1, and y = 0 is revolved about the line x = 1.

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y² = x³, x = 4, and y = 0 is revolved about the y axis.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = (7 - x)², x = 7, and y = 1 is revolved about the line x = 7.

Use the method of washers to find the volume of the solid that results when the area enclosed by y² = 16x, y = 5, and x = 7 is revolved about the x axis.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x³, x = 1, and y = -8 is revolved about the line y = -1.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x⁴, x = 0, and y = 3 is revolved about the y axis.

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.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y² = x³, x = 3, and y = 0 is revolved about the x axis.

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.

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y² = 9x, x = 16, and y = 0 is revolved about the y axis.

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 the method of washers to find the volume of the solid that results when the area of the region enclosed by y = x², y = 1, and x = 0 is revolved about the x axis.

Answer true or false: A right-circular cylinder of radius 8cm contains a hollow sphere of radius 7 cm. If the cylinder is filled to a height h with water and the sphere floats so that its highest point is 4cm above the water level, there is cm³ of water in the cylinder.

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves x = y², x = y + 20, about the y-axis. Approximate area to three decimal places.

Answer true or false. The volume of the solid that results when the region enclosed by the curves x = y², x = y⁵ is revolved about the y-axis is given by .

Find the volume of the hemisphere created by rotating the area enclosed by the circle x² + y² = 4 and above y = 0, around the y-axis. Approximate area to three decimal places.