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8-62.

The figure at right shows the region bounded by the $x$-axis, $f\left(x\right) = 0.5x^{2}$, $x = 1$, and $x = 3$ . The region is revolved around the y-axis to create the solid shown with dotted lines.

1. Describe a method you can use to find the volume.

Will you use Washers or Disks? Will the bounds and integrand be written in terms of $x$ or $y$?

2. Setup the integrals and find the volume. (Using washers, the solution will require two integrals.)

The outside of this solid will be a cylinder with radius $3$.

The inside of the solid will have a hole. But what shape is that hole? Is it cylindrical?

$\text{Or is it determined by }f(y) = \sqrt{(2y)}?$

Notice that the bottom of the hole is cylindrical (with radius $1$), while the top of the hole is determined by $f(y) = \sqrt{(2y)}.$

Consequently, you will need to use the Washer Method twice... both of which are rotated about the y-axis.

Use the eTool below to help solve the problem.
Click on the link to the right to view the full version of the eTool. Calc 8-62 HW eTool