Shell Example

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Situation: You must find the volume of the solid of revolution formed by revolving the region bounded by the graph of x = e^-y2 and the y-axis (0 < y < 1) about the x-axis.

Solution: Since the axis of revolution is horizontal, use a horizontal representative rectangle, as shown below. The width ¤y indicates that y is the variable of integration. The distance from the center of the rectangle to the axis of revolution is p(y) = y, and the height of the rectangle is h(y) = e^-y2. Because the y ranges from 0 to 1, the volume of the solid is
               _{d                               1}
V = 2pi §  p(y)h(y) dy = 2pi §  ye^-y2 dy
               ^{c                               0}
                  ₁
= [-pi e^-y2]
                  ⁰
= pi(1 - 1/e)

= 1.986