# Volume of a Solid by Revolution

1. Feb 10, 2007

### Sidew

1. The problem statement, all variables and given/known data
Hi, the next problem I thought it was easy...and I really think it is, but I haven't come with the right answer :S. I compare the answer in an internet page of problems and all it says is "wrong"...

The problem is the following:
Find the volume of the solid obtained by rotating the region bounded by

2. Relevant equations
I made the sketh in Maple:
http://img109.imageshack.us/img109/5669/solidvolumexi5.th.jpg [Broken]

3. The attempt at a solution
Then I solve for x:

x=(y/8)-4
and wrote the Integral with respect to y, and limits from 0 to 32 (where the function y=8x+32 intersects the y-axis).
Integral of Pi*f(x)^2, from y=0 to y=32, it should give me the volume, isn't it?

http://img164.imageshack.us/img164/98/equationie6.th.jpg [Broken]
My answer: 512/3 * Pi= 536.1651462

But the internet page says I'm wrong but I don't know why :S

Any help is welcome, Thanks in advance!

Last edited by a moderator: May 2, 2017
2. Feb 10, 2007

### sphoenixee

Your answer looks right. I'd say the super trustworthy internet is wrong ;). (Disclaimer: I'm getting back into calc after a year of none, but still I'm quite sure you're correct).

3. Feb 10, 2007

### AngeloG

Possibly: Integration of Pi * (8x + 32)^2. -4 (lower limit) and 0 (upper limit)

Last edited: Feb 10, 2007
4. Feb 10, 2007

### Sidew

AngeloG, I already tried what you suggest and I came with the answer:
4289.321169 which is wrong as well.

And because the region should be rotated about the Y-Axis and not about the X-axis, that's why I solved the equation for X, in order to have only Y's so I could integrate with respect to dy, then my limits are Y=0 and Y=32.

I hope I didn't understand you another thing, maybe "4289.321169" wasn't what you get, so I would like to know your answer.

Thx again^^, cheers!

5. Feb 10, 2007

### AngeloG

Err, forgot to say change the x's for y's and the y's for x's.

8x + 32 is basically the same as 8y + 32. One is just tilted on it's side, which is the y. Then you can integrate from -4 to 0, then rotate it around y axis tilted.

(4096 / 3) * Pi, which is ~4289.3

However, I am wrong =). I was just doing the other half, considering your half was wrong. There's only two ways to do this and if both provide the wrong answer. Might be the page.

Last edited: Feb 10, 2007
6. Feb 11, 2007

### HallsofIvy

Staff Emeritus
y= 8x+ 32 is a straight line with intercepts (0, 32) and (-4,0), rotating around the y-axis gives a cone with height 32 and radius 4. you can check the volume by using the standard formula for volume of a cone: $V= \frac{1}{3}\pi r^2h$.

7. Feb 11, 2007

### Sidew

Thx again for all your answers^^, I didnt know there will be so many options or ways to solve the problem.