Part 2 of 2
Let R be the region bounded by , and . Use the shell method to find the volume of the solid
generated when R is revolved about the line .
Set up the integral that gives the volume of the solid using the shell method. Select the correct choice below and fill
in the answer boxes to complete your choice.
(Type exact answers.)
A.
B.
The volume of the solid is
(Type an exact answer.)

Ask by Graham Wood. in the United States

Mar 29,2025

Question

Part 2 of 2
Let R be the region bounded by , and . Use the shell method to find the volume of the solid
generated when R is revolved about the line .
Set up the integral that gives the volume of the solid using the shell method. Select the correct choice below and fill
in the answer boxes to complete your choice.
(Type exact answers.)
A.
B.
The volume of the solid is
(Type an exact answer.)

Ask by Graham Wood. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

To find the volume of the solid generated when the region R is revolved about the line $$ x = 16 $$ using the shell method, we need to set up the integral that gives the volume of the solid.

The shell method formula for finding the volume of a solid is given by:
$$ V = 2\pi \int_{a}^{b} r(x)h(x) \, dx $$
where:
- $$ r(x) $$ is the radius of the shell,
- $$ h(x) $$ is the height of the shell,
- $$ a $$ and $$ b $$ are the limits of integration.

In this case, the region R is bounded by $$ y = x^2 $$, $$ x = 1 $$, and $$ y = 0 $$. When revolved about the line $$ x = 16 $$, the radius of the shell is $$ 16 - x $$ and the height of the shell is $$ x^2 $$.

Therefore, the integral that gives the volume of the solid using the shell method is:
$$ V = 2\pi \int_{0}^{1} (16 - x)x^2 \, dx $$

So, the correct choice is:
B. $$ \int_{0}^{1} (2\pi(16 - x)x^2) \, dx $$
The volume of the solid is $$ \int_{0}^{1} 2\pi(16 - x)x^2 \, dx $$.

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