In Problems 11-16, use the disk method to find the volume of...

In Problems $11-16,$ use the disk method to find the volume of the solid of revolution generated by revolving the region bounded by the graphs of the given equations about the indicated axis.
$$ y=x^{2}, x \geq 0, y=1, y=4 ; \text { about the } y \text { -axis } $$

Key Concepts

Solid of Revolution

A solid of revolution is a three-dimensional shape obtained by revolving a two-dimensional region about an axis. In many calculus problems, the given region is defined by one or more curves and straight-line boundaries, and rotating this region around an axis creates a solid whose volume can be determined using integration techniques like the disk method.

Setting Up the Integral

Setting up the integral involves identifying the correct variable of integration, expressing the radius of the disks (or washers) in terms of this variable, and determining the limits of integration from the boundaries of the region. When rotating about the y-axis, it is often necessary to solve for x as a function of y, thereby expressing the radius of each disk in terms of y.

Disk Method

The disk method is a technique in integral calculus used to find the volume of a solid of revolution. It involves slicing the solid perpendicular to the axis of rotation into thin, flat disks, finding the area of each disk, and then integrating these areas over a given interval. This method is particularly useful when the slices are circular and have no hole, allowing the volume to be computed as an integral of the form V = ??(R(y))^2 dy or V = ??(R(x))^2 dx.

Changing Variables and Finding Limits

In problems where the region is defined using an equation such as y = x^2 and the solid is generated by rotation about the y-axis, one must convert the function by solving for x in terms of y (x = ?y for x ? 0). This transformation is crucial because it allows you to correctly set up the integral with the variable that corresponds to the axis of revolution, and determine the corresponding integration limits based on the given boundaries.

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