# Volume of a Solid by Revolution

## Homework Statement

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

## Homework Equations

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

## 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!

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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).

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

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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!

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.

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HallsofIvy
Homework Helper
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: $V= \frac{1}{3}\pi r^2h$.

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