Use the shell method to find the volume of the following...

Use the shell method to find the volume of the following solids.
The solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cylinder of height 6 and radius 4

Key Concepts

Shell Method

The shell method is an integration technique used to find the volume of a solid of revolution. Instead of slicing the solid into washers or discs, the region is decomposed into cylindrical shells. Each shell has a small thickness and can be thought of as a curved rectangle wrapped around the axis of rotation. The volume is then computed by summing up (integrating) the contribution of each of these shells over the given interval.

Volumes of Solids of Revolution

Volumes of solids of revolution are calculated by integrating cross-sectional areas as a figure is rotated around an axis. The method of cylindrical shells is one of several techniques (others include the disc and washer methods) that facilitate the calculation by exploiting the symmetry of rotation to set up an appropriate integral that represents the entire volume.

Setting Up an Integral Using Cylindrical Shells

When using the shell method, the key steps involve identifying the radius, height, and thickness of a typical shell. The radius is the distance from the axis of rotation to the shell, the height is determined by the function defining the shape, and the thickness is an infinitesimal change in the variable. These components are combined in the expression for the volume of a thin shell, and integration over the appropriate bounds sums the volume of all such shells to obtain the total volume of the solid.

Composite Solids and Volume Subtraction

In problems where a portion of a solid is removed, such as drilling a hole through a body, the total volume can be computed by subtracting the volume of the removed part from the volume of the original solid. Understanding this method of composing or subtracting volumes is crucial in applying the shell method to composite solids, ensuring that only the volume of the remaining body is considered.

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