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

Verified step by step guidance

1. 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.
2. 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).
3. 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: \(r(y) = 3 \left(1 - \frac{y}{8}\right)\).
4. Write the formula for the volume using the shell method. The volume is given by integrating the lateral surface area of cylindrical shells: \(V = \int_0^8 2\pi \cdot r(y) \cdot \text{height of shell} \ dy\). 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 \(dy\).
5. 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.

Key Concepts

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. Each shell's volume is approximated by its circumference times height times thickness, and the integral sums these volumes over the given interval.

Setting up the Integral for a Cone

To find the volume of a cone using the shell method, express the radius and height of each shell in terms of the variable of integration. For a cone, the radius changes linearly with height, so identifying this relationship is key to setting up the integral correctly.

Geometry of a Right Circular Cone

A right circular cone has a circular base and a vertex aligned above the center of the base. Its radius and height define its shape, and understanding how these dimensions relate helps in modeling the solid for volume calculations using calculus methods.