Consider the region R given in Exercise 63 . If the volume...

Consider the region $R$ given in Exercise $63 .$ If the volume of the
solid formed by revolving $R$ around the $x$ -axis is $6 \pi,$ and the vol-
ume of the solid formed by revolving $R$ around the line $y=-2$ is
$10 \pi,$ find the area of $R .$

Key Concepts

Pappus's Centroid Theorem

Pappus's Centroid Theorem (also known as the Guldinus theorem) provides a relation between the volume of a solid of revolution and the area of the plane region being rotated. Specifically, it states that the volume generated by revolving a plane region about an external axis in its plane is equal to the product of the area of the region and the distance traveled by its centroid during the rotation (which is typically 2? times the distance from the centroid to the axis). This theorem is especially useful for linking the given volumes of rotated regions to the area and centroid of the original region.

Solid of Revolution

A solid of revolution is generated when a plane region is rotated around a given axis. The standard methods for finding its volume are the disk/washer method and the shell method, which involve integrating cross-sectional areas or cylindrical shells. In many problems, knowing the relationship between the volume of the solid and the geometry of the region is key to setting up equations that relate the area of the region to given volumes.

Centroid of a Plane Region

The centroid (or geometric center) of a plane region is the average location of all the points in the region, and its coordinates can be computed as the weighted average of the region's coordinates. In problems involving rotation, the distance from the centroid to the axis of rotation often plays a crucial role in determining properties such as volume or surface area of the resulting solid.

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