Integration By Parts?

[normabeatty]

Hello. The topic in class atm is IBP.

We were given an assignment of 6 questions. ill post the others i cant answer, with my work, here.

4. use cylindrical shells. find volume bounded by:
y = e^x, y = e^-x, and x = 1 about the y-axis

what I did was follow the preset formula for cylind shells.
v = int(from a to b) (2pix)(f(x))dx
so...

h = e^x - e^-x
r = 1 - x

so...

v = 2pi int(from 0 to 1) ((e^x - e^-x)(1-x))dx
And this is where i get stuck.
I don't know where to apply the IBP. I don't think I can solve it with substitution.
If I expand it....that's:

v = 2pi int(from 0 to 1) (e^x - xe^x - e^-x + x^-x)dx

where to go from here?

Thank you. And what happened to the live chat thing?

 

[HallsofIvy]

Hello. The topic in class atm is IBP.

We were given an assignment of 6 questions. ill post the others i cant answer, with my work, here.

4. use cylindrical shells. find volume bounded by:
y = e^x, y = e^-x, and x = 1 about the y-axis

what I did was follow the preset formula for cylind shells.
v = int(from a to b) (2pix)(f(x))dx
so...

h = e^x - e^-x
r = 1 - x

so...

v = 2pi int(from 0 to 1) ((e^x - e^-x)(1-x))dx
And this is where i get stuck.
I don't know where to apply the IBP. I don't think I can solve it with substitution.
If I expand it....that's:

v = 2pi int(from 0 to 1) (e^x - xe^x - e^-x + x^-x)dx

Separate those integrals. \(\displaystyle 2\pi\int_0^1 e^x dx- 2\pi\int_0^1 xe^x dx- 2\pi\int_0^1 e^{-x} dx+ 2\pi\int_0^1xe^{-x}dx\)
(I take it that \(\displaystyle x^x\) on the end was a typo.)
\(\displaystyle e^x\) and \(\displaystyle e^{-x}\) are direct anti-derivatives. \(\displaystyle xe^x\) and \(\displaystyle xe^{-x}\) can be integrated by parts.

where to go from here?

Thank you. And what happened to the live chat thing?

 

[DrPhil]

Thank you. And what happened to the live chat thing?

I hope I am not stepping on anybody's toes here, but the only free live chat I know of is through a different site:

http://www.pathwhelp.org

Hours are 4-10 Eastern time, Sunday through Thursday.
Business over there has fallen so low that I spend all of my "Live Room" time working forum problems over here!