31. Find the volume of the solid generated when R (shaded...

31. Find the volume of the solid generated when R (shaded region) is revolved about the given line. y = 12 - \sqrt{x}, y = 12, x = 144; about y = 12 The volume of the solid obtained by revolving the region about y = 12 is (Type an exact answer, using \pi as needed.)

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31. Find the volume of the solid generated when the shaded region is revolved about the given line. Given: Region: Ay Q y = 12 - x, y = 12, x = 144 Line of revolution: y = 12 To find the volume of the solid obtained by revolving the region about y = 12, we can use the method of cylindrical shells. The height of each cylindrical shell is given by the difference between the upper and lower curves: (12 - x) - 12 = -x. The radius of each cylindrical shell is the distance from the line of revolution (y = 12) to the curve y = 12 - x. This distance is given by the equation: 12 - (12 - x) = x. The differential volume of each cylindrical shell is given by the formula: dV = 2πrh dx, where r is the radius and h is the height. Therefore, the volume of the solid is given by the integral of the differential volume from x = 144 to x = 145: V = ∫[144,145] 2πx(-x) dx Now, we can solve the integral to find the volume of the solid.

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31. Find the volume of the solid generated when R (shaded region) is revolved about the given line. y = 12 - \sqrt{x}, y = 12, x = 144; about y = 12 The volume of the solid obtained by revolving the region about y = 12 is (Type an exact answer, using \pi as needed.)

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