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 curve y = 1+√x, the curve y = 1−√x, and the line x=1 is revolved about the y-axis. Find the volume of the resulting solid by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your answers agree.

Accepted Answer

First, understand the region bounded by the curves: $$y = 1 + \sqrt{x}$$, $$y = 1 - \sqrt{x}$$, and the vertical line $$x = 1. $$Sketching the region helps visualize the area to be revolved around the y-axis. For part (a), integrating with respect to $$x$$: Since the solid is formed by revolving around the y-axis, use the shell method. The shell radius is the distance from the y-axis, which is $$x$$, and the shell height is the difference between the upper and lower curves, $$\left( (1 + \sqrt{x}) - (1 - \sqrt{x}) ight) = 2\sqrt{x}. $$The volume integral is then $$V = \$$int_0$$^1 2\pi \cdot (	ext{radius}) \cdot (	ext{height}) \, dx. $$Set up the integral for part (a): $$V = \$$int_0$$^1 2\pi x \cdot 2\sqrt{x} \, dx = \$$int_0$$^1 4\pi x \sqrt{x} \, dx. $$Simplify the integrand before integrating. For part (b), integrating with respect to $$y$$: Express $$x in $$terms of $$y$$ from the given curves. From $$y = 1 + \sqrt{x}$$, solve for $$x to $$get $$x = (y - 1)^2$. Similarly, from y = 1$$ - \sqrt{x}$$, x = (1$$ - $$y)^2. $$The region in $$y$$ goes from $$y = 0 to$$ $$y = 2 ($$since at $$x=1$$, $$y$$ ranges between $$1 - 1 = 0$$ and $$1 + 1 = 2). $$Use the disk/washer method for part (b): The radius of the washers is the horizontal distance from the y-axis to the outer curve minus the inner curve, which is $$R(y) = 1 ($$since $$x=1 is $$the boundary) and $$r(y) = (1 - y)^2$ or (y$$ - $$1)^2$$ depending on the side. Set up the volume integral as $$V = \$$int_0$$^2 \pi \left( R(y)^2$ - r(y)^2$ ight) dy. $$Evaluate this integral to find the volume.

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Concetti chiave

Volume of Solids of Revolution

Disk/Washer and Shell Methods

Integration with Respect to Different Variables