Let R be the region bounded by the upper half of the circle...

Let $R$ be the region bounded by the upper half of the circle $x^{2}+y^{2}=r^{2}$ and the $x$ -axis. A sphere of radius $r$ is obtained by revolving $R$ about the $x$ -axis. a. Use the shell method to verify that the volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^{3}$ b. Repeat part (a) using the disk method.

Let $R$ be the region bounded by the upper half of the circle $x^{2}+y^{2}=r^{2}$ and the $x$ -axis. A sphere of radius $r$ is obtained by revolving $R$ about the $x$ -axis.
a. Use the shell method to verify that the volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^{3}$
b. Repeat part (a) using the disk method.

Key Concepts

Disk Method

The disk method calculates the volume of a solid of revolution by slicing the solid perpendicular to the axis of rotation and summing the volumes of the resulting circular disks (or washers, when there is a hole in the middle). Each disk’s volume is determined by its cross-sectional area multiplied by its thickness. This method is most straightforward when the cross-sectional areas are easily expressible as functions of the variable along the axis of rotation.

Integration Technique

Integration is the mathematical process of adding infinitely small quantities to obtain a complete total, such as area, volume, or other accumulated quantities. In the context of volumes of revolution, setting up the proper integral expression is crucial, as it involves identifying the correct limits of integration and the appropriate integrand representing the volume of a shell or disk. This forms the backbone of the calculus-based methods for determining volumes.

Solids of Revolution

This concept involves creating a three-dimensional object by rotating a two-dimensional region around an axis. The resulting volume is typically computed by slicing the solid into small pieces and summing their individual volumes through integration. This idea is central in calculus for determining volumes of irregular or curved shapes, and it forms the foundation for both the shell and disk methods.

Shell Method

The shell method is a technique for finding the volume of a solid of revolution by integrating along the axis perpendicular to the axis of rotation. It involves summing up the volumes of cylindrical shells, where each shell’s volume is given by the product of its circumference, height, and thickness. This method is especially useful when the solid is generated by rotating a region that is easier to describe in terms of the variable perpendicular to the axis of rotation.

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