# Thread: Volume of region bounded by two graphs?

1. ## Volume of region bounded by two graphs?

The region is bounded by y=x^3 and y=x in the first quadrant.

I already found the area, but I'm having trouble with the volume.

First:
A)find the volume rotated around y=2
B) find volume rotated around the y axis
C)the area (1/4) is the base of a solid. There are cross sections perpendicular to the x-axis that are semi circles. Find the volume of the solid.

Help?

2. The formula to find the volume is:

$
\pi \int (outer - line going around)^2 - (inner - line going around)^2
$

That means "a" would be setup as follows:

$
2 \times \pi \int (x-2)^2 - (x^3-2)^2
$

The integral will be from 0 to 1.

The reason we multiply it by 2 is because the two lines actually form a solid twice. Therefore we will take the integral from a point of intersection to the next point of intersection on one solid and multiply times 2 in order to get the volume of both solids.

b)To find the area going around the y-axis we need to place all of our $x$'s in terms of $y$.

We come out with $x=\sqrt[3]{y}$ and $x=y$. Remember the graph looks the same, but the equations are just in terms of $y$.

Then just use the formula to find the volume.

Your setup should look as follows:

$2 \times \pi \int (\sqrt[3]{y})^2 - (y)^2$

The integral will be from 0 to 1 again because the graphs intersect at those points.

Question "c" is beyond my knowledge. Maybe someone else here can enlighten me.

If you have any questions, just ask!

3. So, would the graph be considered dx or dy? Because oringally I thought it was dx, but then it asked for it to be rotated around y=2 and the y-axis. So, in order to do that, don't I need to make the y=2 volume change from x to y like in (b)?

4. No, for "a" your terms need to be in terms of $x$.

This is because $y=2$ is virtually the same as the $x-axis$, but shifted up 2.

5. ok. And it's the washer method because of the formula used? Btw, I appreciate your help!

6. You got it!

No problem and good luck!

7. Ok. That's probably what I was doing wrong. I had thought it was the shell method because it made a "cup" shape. Haha.