53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x² and y = 2−x²; about the x-axis

Verified step by step guidance

1. First, identify the region R bounded by the curves \(y = x^{2}\) and \(y = 2 - x^{2}\). To do this, find the points of intersection by setting \(x^{2} = 2 - x^{2}\) and solving for \(x\).
2. Solve the equation \(x^{2} = 2 - x^{2}\) to find the limits of integration. This will give you the \(x\)-values where the two curves intersect, which define the interval over which the volume will be calculated.
3. Since the solid is generated by revolving the region about the x-axis, use the washer method. The volume element is given by the difference of the areas of two disks: \(V = \pi \int_{a}^{b} \left(R_{outer}^{2} - R_{inner}^{2}\right) \, dx\).
4. Determine the outer radius \(R_{outer}\) and inner radius \(R_{inner}\) for the washers. Because the region is between \(y = x^{2}\) and \(y = 2 - x^{2}\), and the axis of rotation is the x-axis, the radii correspond to the distances from the x-axis to each curve: \(R_{outer} = 2 - x^{2}\) and \(R_{inner} = x^{2}\).
5. Set up the integral for the volume: \(V = \pi \int_{a}^{b} \left[(2 - x^{2})^{2} - (x^{2})^{2}\right] \, dx\), where \(a\) and \(b\) are the intersection points found in step 2. This integral can then be evaluated to find the volume.

Key Concepts

Finding the Region Bounded by Curves

To find the volume of a solid generated by revolving a region, first identify the area bounded by the given curves. This involves determining the points of intersection and understanding which curve lies above or below within the interval. For y = x² and y = 2 − x², solving for their intersection points helps define the limits of integration.

Method of Disks/Washers for Volume Calculation

The disk or washer method calculates volume by slicing the solid perpendicular to the axis of revolution. Each slice forms a disk or washer whose volume is π times the difference of the outer and inner radii squared, integrated over the interval. When revolving around the x-axis, the radii are expressed as functions of x.

Setting Up and Evaluating Definite Integrals

After determining the radii and limits, set up a definite integral representing the volume. The integral sums the volumes of infinitesimally thin disks or washers across the interval. Evaluating this integral yields the total volume of the solid formed by revolving the region about the axis.