Calculating Volume with the Shell Method | A Technique for Finding Volumes of Solids of Revolution in Calculus

Shell Method

The shell method is a technique used in calculus to calculate the volume of a solid of revolution

The shell method is a technique used in calculus to calculate the volume of a solid of revolution. It is typically used when rotating a two-dimensional shape around an axis to form a three-dimensional solid.

To understand the shell method, let’s imagine a simple example. Suppose we have a region bounded by the graph of a function f(x), the x-axis, and two vertical lines x = a and x = b. We want to rotate this region around the y-axis to create a solid.

To find the volume of this solid using the shell method, we divide the region into infinitely thin vertical strips. Each strip has a width Δx and a height given by the difference between the top and bottom points of the strip, which can be represented as (f(x) – g(x)), where g(x) represents the lower boundary of the strip.

Now, if we think of each strip as a shell, we can imagine stacking these shells next to each other along the y-axis to create the solid. The volume of each shell can be approximated by multiplying its height (f(x) – g(x)) by the circumference of the shell, which is given by 2πx (since the circumference of a circle is 2πr, and in this case, x represents the radius).

By summing up the volumes of all the infinitely thin shells, we can obtain an integral expression for the total volume of the solid:

V = ∫[a,b] (2πx)⋅(f(x) – g(x)) dx

This integral can be evaluated to find the exact volume of the solid.

In summary, the shell method is a technique used in calculus to find the volume of a solid of revolution by considering the infinitesimally thin shells formed when rotating a region around an axis. It provides an alternative approach to the disc method for calculating volumes of solids.

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