43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?

Verified step by step guidance

1. First, identify the region bounded by the curves: \(y = 2x\), \(y = 6 - x\), and \(y = 0\). Sketching or analyzing these will help visualize the area to be revolved.
2. Determine the points of intersection between \(y = 2x\) and \(y = 6 - x\) by setting \$2x = 6 - x\( and solving for \)x$. This gives the limits of integration for the volume calculation.
3. For the solid revolved about the line \(y = -2\), use the disk/washer method because the axis of rotation is horizontal and outside the region. Express the radius of each disk/washer as the distance from the curve to the line \(y = -2\).
4. Set up the integral for the volume about \(y = -2\) by integrating with respect to \(x\) between the intersection points. The volume element is \(\pi \times (\text{outer radius}^2 - \text{inner radius}^2) \, dx\), where the radii are distances from the curves to \(y = -2\).
5. For the solid revolved about the line \(x = -2\), use the shell method because the axis of rotation is vertical and outside the region. Express the height of each shell as the difference between the two curves, and the radius as the distance from \(x\) to \(-2\). Set up the integral with respect to \(x\) using the formula \(2\pi \times (\text{radius}) \times (\text{height}) \, dx\) over the same interval.

Key Concepts

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around a line (axis). The volume is computed by integrating cross-sectional areas perpendicular to the axis of rotation, using methods like disks, washers, or shells.

Disk/Washer Method

The disk/washer method calculates volume by slicing the solid perpendicular to the axis of rotation, forming circular cross-sections (disks) or rings (washers). The volume is found by integrating the area of these disks or washers along the axis.

Shell Method

The shell method involves slicing the solid parallel to the axis of rotation, creating cylindrical shells. The volume is computed by integrating the lateral surface area of these shells, which is useful when the disk/washer method is complicated.