Find the volume of the solid generated when the indicated plane region is revolved about

the axis of revolution. The region bounded by y = x

2 − 2x + 1, y = 7 − x about y = −1.

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This answer was edited.The sketch of the solid formed by the revolving the region bounded by and about is shown in the attached figure.

From the figure, if we slice out tiny pieces of the solid, we will have cylinders with a height of , and the base area of , where is the radius of the outer shell minus the radius of the inner shell. Notice that the outer shell is made by the line and because the axis of revolution is , the radius of the outer shell (distance from the axis of revolution) is given by . Similarly, the inner shell is made by the curve and because the axis of revolution is , the radius of the inner shell (distance from the axis of revolution) is given by .

Thus, the base area of each slice of cylinder is given by .

.

Thus, the volume of the solid formed by revolving the region bounded by and about is given by

cubic units.