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 graph of y = 4−x² and the x-axis on the interval [−2,2] is revolved about the line x = −2. What is the volume of the solid that is generated?

Verified step by step guidance

1. First, visualize the region bounded by the curve \(y = 4 - x^{2}\) and the x-axis on the interval \([-2, 2]\). This region lies above the x-axis and below the parabola between \(x = -2\) and \(x = 2\).
2. Since the solid is generated by revolving this region about the vertical line \(x = -2\), consider using the shell method because the axis of rotation is vertical and the slices perpendicular to the x-axis will be easier to handle.
3. Set up a representative cylindrical shell at a position \(x\) between \(-2\) and \$2\(. The height of the shell is given by the function value \)h(x) = 4 - x^{2}\(, and the radius of the shell is the horizontal distance from \)x\( to the line \)x = -2\(, which is \)r(x) = x - (-2) = x + 2$.
4. The volume of each thin shell is given by the formula \(dV = 2\pi \times \text{radius} \times \text{height} \times \text{thickness} = 2\pi (x + 2)(4 - x^{2}) \, dx\).
5. To find the total volume, integrate the expression for \(dV\) over the interval \(x = -2\) to \(x = 2\): \(V = \int_{-2}^{2} 2\pi (x + 2)(4 - x^{2}) \, dx\). This integral can then be evaluated to find the volume.

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 of revolution). The volume is computed by integrating cross-sectional areas perpendicular to the axis, using methods like disks, washers, or shells depending on the axis and shape.

Shell Method

The shell method calculates volume by summing cylindrical shells formed by revolving vertical or horizontal slices around an axis. Each shell's volume is 2π(radius)(height)(thickness), making it useful when the axis of rotation is parallel to the slices and not the coordinate axis.

Setting up the Integral with Correct Limits and Radius

Accurately determining the radius and height of slices relative to the axis of rotation is crucial. For rotation about x = -2, the radius is the horizontal distance from x to -2, and the height is given by the function y = 4 - x². Proper limits correspond to the interval of the region.