Consider the region R given in Exercise 49 . If the area of...

Consider the region $R$ given in Exercise $49 .$ If the area of region $R$ is $1,$ and the volume of the solid formed by revolving $R$ about the line $x=-3$ is $10 \pi,$ find the volume of the solid formed by revolving $R$ about the $y$ -axis.

Key Concepts

Axis of Rotation

The axis of rotation is the line about which the region is revolved to form the solid. Its position relative to the region significantly affects the volume of the generated solid because it changes the distance the centroid travels during the rotation. Shifting the axis of rotation typically alters the integral setup or the application of Pappus's theorem, making it a key factor in analyzing and solving these problems.

Centroid of a Plane Region

The centroid of a plane region represents its geometric center or balance point. When a region is revolved about an axis, the distance from the centroid to the axis determines the path length the centroid travels, which is critical in applying Pappus's Centroid Theorem. Knowing the location of the centroid allows one to relate properties of the region (such as its area) to the volume of the resulting solid of revolution.

Solid of Revolution

This concept involves forming a three?dimensional object by rotating a two-dimensional region about an axis. It is a fundamental idea in calculus for calculating volumes of such objects, either through integration methods like the disk/washer or shell method, or through geometric theorems. This approach provides a structured way to decompose and accumulate infinitesimal volumes to arrive at the total volume of the solid.

Pappus's Centroid Theorem

Pappus's Centroid Theorem relates the volume of a solid of revolution to the area of the region being revolved and the distance traveled by its centroid. Specifically, the theorem states that the volume equals the product of the area of the shape and the distance that its centroid travels during the rotation (i.e., the circumference of the circular path). This theorem simplifies volume calculations, especially when the centroid's position can be determined or inferred.

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