Question

use the shell method to write and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.x y2 = 16

Answer 1

The volume of the generated solid is found using the shell method by setting up an integral that multiplies the circumference of cylindrical shells by their height and thickness, and integrating from the lower to upper bounds of x.

Step-by-step explanation:

To use the shell method to find the volume of the solid generated by revolving the region around the x-axis where the equation is x = y2/16, we first rearrange the equation as y = ±4√x.

We treat y as the radius of a cylindrical shell with thickness dx and height y.

The volume of the shell is then the circumference (2πy) times the height (y) times the thickness (dx). The volume of the entire solid is calculated by integrating these volumes from x = 0 to x = 16.

The integral for the volume is: V = ∫0162πy(y)dx. Substituting y = 4√x, we have V = ∫0162π(4√x)(4√x)dx = ∫01632πx dx. Finally, we evaluate the integral to find the volume.