Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

The region bounded by the graph of y = 4−x² and the x-axis on the interval [−2,2] is revolved about the line x = −2. What is the volume of the solid that is generated?

Accepted Answer

First, visualize the region bounded by the curve $$y = 4 - x$$^{2}$$ and the x-axis on the interval $$[-2, 2]$$. This region lies above the x-axis and below the parabola between x = -2$ and x = 2$. Since the solid is generated by revolving this region about the vertical line x = -2$, consider using the shell method because the axis of rotation is vertical and the slices perpendicular to the x-axis will be easier to handle. Set up a representative cylindrical shell at a position x$ between -2$ and 2$. The height of the shell is given by the function value h(x) = 4$$ - $$x^{2}$$, and the radius of the shell is the horizontal distance from $$x to $$the line $$x = -2$$, which is $$r(x) = x - (-2) = x + 2. $$The volume of each thin shell is given by the formula $$dV = 2\pi 	imes 	ext{radius} 	imes 	ext{height} 	imes 	ext{thickness} = 2\pi (x + 2)(4 - x$$^{2}$$) \, dx. To $$find the total volume, integrate the expression for $$dV$$ over the interval $$x = -2 to$$ $$x = 2$$: $$V = \int_{-2}^{2} 2\pi (x + 2)(4 - x$$^{2}$$) \, dx. $$This integral can then be evaluated to find the volume.

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Key Concepts

Volume of Solids of Revolution

Shell Method

Setting up the Integral with Correct Limits and Radius