Question

64–68. Shell method Use the shell method to find the volume of the following solids.A right circular cone of radius 3 and height 8

Accepted Answer

Identify the axis of rotation and the shape of the solid. Here, we are dealing with a right circular cone of radius 3 and height 8, and we want to find its volume using the shell method. Set up a coordinate system to describe the cone. For example, place the cone so that its height extends along the y-axis from 0 to 8, and the radius extends along the x-axis from 0 to 3 at the base (y = 0). Express the radius of the shell at a given height y. Since the cone tapers linearly, the radius r(y) at height y can be found using similar triangles: $$r(y) = 3 \left(1 - \frac{y}{8}ight). $$Write the formula for the volume using the shell method. The volume is given by integrating the lateral surface area of cylindrical shells: $$V = \$$int_0$$^8 2\pi \cdot r(y) \cdot 	ext{height of shell} \ dy. $$Here, the height of each shell corresponds to the horizontal distance, which is $$x$$, but since we are integrating with respect to y, the shell height is the radius $$r(y)$$ and the shell thickness is $$dy. $$Set up the integral explicitly: $$V = \$$int_0$$^8 2\pi \cdot r(y) \cdot y \ dy$$, where $$y is $$the height of the shell and $$r(y) is $$the radius from step 3. Then, substitute $$r(y)$$ and prepare to evaluate the integral to find the volume.

64–68. Shell method Use the shell method to find the volume of the following solids.

A right circular cone of radius 3 and height 8

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Key Concepts

Shell Method for Volume

Setting up the Integral for a Cone

Geometry of a Right Circular Cone