Use the disk or washer method to find the volume of the...

Use the disk or washer method to find the volume of the solid of revolution that is formed by revolving the region bounded by the graphs of the given equations about the indicated line or axis.
$y=1-x^{2}, y=x^{2}-1, x=0,$ first quadrant; $\quad y$ -axis

Key Concepts

Solid of Revolution

A solid of revolution is created when a two-dimensional region is rotated about an axis. This process generates a three-dimensional object, and finding the volume of such an object typically involves integrating the areas of cross-sectional slices (disks or washers) perpendicular to the axis of rotation.

Disk/Washer Method

The disk method involves slicing the solid into thin disks perpendicular to the axis of rotation, where the volume of each slice is approximated by the area of the circular face times a small thickness. When the solid has a hollow section, the washer method is used, which subtracts the inner radius from the outer radius in the area calculation. Both methods convert the problem of finding a volume into evaluating an integral with respect to a chosen variable.

Bounded Region and Intersection Points

Understanding the bounded region is essential since it provides the limits for integration. It is determined by the curves or lines that define the region, and in many problems, finding the intersection points of these curves is necessary to correctly establish the endpoints of the integral.

Axis of Revolution

The axis of revolution is the line about which the region is rotated to form the solid. Identifying the correct axis is crucial because it affects the radii used in the disk or washer method. The distance from a curve to the axis of rotation determines the element's radius, which is then used in the volume integral.

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