A cone by two methods Verify that the volume of a right...

A cone by two methods 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 around 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 $y$

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Key Concepts

Disk Method

The disk method involves slicing the solid perpendicular to the axis of rotation, producing circular disks (or washers, if there is a hole). Each disk has a volume that is determined by its radius, and integrating these individual volumes over the specified interval yields the total volume of the solid.

Shell Method

The shell method computes volume by slicing the solid parallel to the axis of rotation to form cylindrical shells. By integrating the volumes of these shells, which depend on the radius and height of each shell, one can determine the total volume of the solid. This method is particularly useful when the function describing the boundary is easier to express in the variable that is integrated.

Solid of Revolution

A solid of revolution is a three-dimensional shape that is created by rotating a two-dimensional region about an axis. Understanding this concept is essential because it allows one to use integration methods to compute volumes of solids formed by such rotations.

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