Volumes of Revolution with e^-x

[RJLiberator]

Homework Statement

Compute the region R in the first quadrant between y=e^(-x), x=0, and y=0. Compute using shells, the volume V of solid around the y-axis.

Homework Equations

Volume =integral of bounds 2pi*radius*height

The Attempt at a Solution

First I drew the graph. This graph really is just a graph of e^(-x).
I then visually rotated it around the y-axis.
This problem seems easy enough to set up:

Volume = Integral between 0 (lower limit) and infinity (upper limit) of 2*pi*x*e^-(x) dx
where x = radius
dx = width
e^(-x) = height

This problem was also easy to integrate using integral substitution first, and then integration by parts one time. The final equation was

V = 2pi[-xe^(-x)-e^(-x)] from 0 to infinity.

After calculating the simple answer is 2pi.My problem: For some reason, when I use this calculator: http://www.wolframalpha.com/widgets/view.jsp?id=1cd73be1e256a7405516501e94e892ac

I get an answer of pi/2.

Am I doing something wrong? Or is my answer/thinking correct?

Thank you.

 

[Dick]

To do this by the method of shells you want to integrate dy, not dx

 

[RJLiberator]

Oh, even if it is around the y-axis? Hm. Let me try this out again. Thank you.

 

[LCKurtz]

No, I think Dick misspoke there. Rotating about the y-axis you do want dx elements for shells. I also get $2\pi$.

 

[Dick]

No, I think Dick misspoke there. Rotating about the y-axis you do want dx elements for shells. I also get $2\pi$.

Yes, I did misspeak and 2pi is correct. pi/2 is correct if you are rotating around the x-axis.

 

[RJLiberator]

Ah, magnificent! So my answer is then verified. This makes me overjoyed.

I tried it the other way and it didn't work out too well, was getting -infinity :p.

Thank you, friends, for assisting me tonight.

 

[Mark44]

First I drew the graph. This graph really is just a graph of e^(-x).
I then visually rotated it around the y-axis.

It's not clear from what you wrote, but you should also draw a sketch of the solid of rotation. In your first graph you should include an incremental area element that will be rotated. In your second graph, you should include a sketch of the shell or disk or whatever. If you do that, you'll have a better chance of getting the integrand right, which in this case it seems that you did.