Verify that the volume of a right circular cone with a base...

Verify that the volume of a right circular cone with a base radius of $r$ and a height of $h$ is $\pi r^{2} h / 3 .$ Use the region bounded by the line $y=r x / h,$ the $x$ -axis, and the line $x=h,$ where the region is rotated about the $x$ -axis. Then (a) use the disk method and integrate with respect to $x,$ and (b) use the shell method and integrate with respect to $y$

Key Concepts

Solid of Revolution

A solid of revolution is a three?dimensional object obtained by rotating a two-dimensional region about a specific axis. This concept is fundamental in calculus for computing volumes of objects that possess symmetry, and it forms the basis for using integration methods such as the disk and shell methods to determine the volume of the generated solid.

Disk Method

The disk method involves slicing the solid perpendicular to the axis of rotation, creating a series of thin disks (or circular cross-sections). The volume of the solid is then approximated by summing the volumes of these disks, which is done by integrating the area of a typical disk along the axis of revolution. This method is particularly useful when the cross-sections of the solid are easily expressed as functions of the integration variable.

Shell Method

The shell method entails partitioning the solid into thin cylindrical shells parallel to the axis of rotation. By calculating the volume of each shell—using the product of its circumference, height, and thickness—and integrating these volumes across the given interval, one can find the total volume of the solid. This method is often advantageous when the region is more naturally described with respect to the variable perpendicular to the axis of rotation.

Definite Integration

Definite integration is the process of integrating a function over a specified interval to accumulate quantities such as area or volume. In the context of volume calculation, it is used to sum the contributions of an infinite collection of infinitesimally thin slices or shells to determine the total volume of a solid, making it an essential tool in applications involving continuous summation.

Volume of a Cone

The volume of a cone, specifically a right circular cone, is a classic result in solid geometry that is derived from integrating cross-sectional areas or using geometric reasoning. The formula, which is one-third of the product of its base area and height, can be verified through both the disk method and the shell method, illustrating the power of integration techniques in deriving known volume formulas from first principles.

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