Sketch a plane region and indicate the axis about which it is revolved so that the resulting solid of revolution (found using the shell method) is given by the integral. 2$\pi$

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The integral given is a shell method, which requires the axis of revolution to be parallel to the axis of integration. In this case, the integral has dx as its differential, so the axis of revolution must be parallel to the y-axis. Therefore, the plane region must be revolved about the y-axis, eliminating choices A and B. The only remaining choices are C and D, but the region in choice D is not symmetric about the x-axis and would require splitting into two parts to be integrated. Therefore, the best choice is C, where the region is symmetric about the x-axis and can be easily integrated using the shell method.

Sketch a Plane Region and Indicate the Axis About Which π\piπ

Sketch a Plane Region and Indicate the Axis About Which $\pi$