1. ## Applications of Intergration

Hey guys I have a quick question.

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell.

For this question would i just sub in 3 into the y and just solve and find the interval. Then do the integration of 2pie x (4x-x^2) between the two intervals?

2. ## Re: Applications of Intergration

Your shells will extend from y = 0 , to y = 4x-x^2 . They will be too high.

The integrand should be f(x) - g(x).

You just have f(x).

3. ## Re: Applications of Intergration

that's all they gave me though, so how do i continue?

4. ## Re: Applications of Intergration

Hello, kashmoneyrecord3!

I don't understand your game plan,
but "just sub in 3" sounds like a bad idea . . .

Use the method of cylindrical shells to find the volume generated by rotating
the region bounded by the given curves about the specified axis.
Sketch the region and a typical shell.

. . $y\:=\:4x-x^2,\; y\,=\,3,\;\text{ about }x=1$

Did you make a sketch?

Code:
        |
|  :   ..*..
|  :.*:::::::*.
3+  *- - - - - -*
| *:           :*
|  :           :
|* :           : *
|  :           :
|  :           :
----*--+-----------+--*----
|  1           3  4
|

Formula: . $V \;=\;2\pi\! \int^b_a\!\text{(radius)(height)}\,dx$

We have: . $\begin{Bmatrix}r \,=\, x-1 \\ h \,=\,(4x-x^2) - 3 \\ a \,=\, 1 \\ b \,=\, 3 \end{Bmatrix}$

Hence: . $V \;=\;2\pi\! \int^3_1\!(x-1)(4x-x^2-3)\,dx$

5. ## Re: Applications of Intergration

how did you get the intervals?

6. ## Re: Applications of Intergration

Originally Posted by kashmoneyrecord3
how did you get the intervals?
Just the way you said in the Original Post, "just sub in 3 into the y".

Solve 3 = 4x-x^2 , for x. That should give x = 1, x = 3