Question

The region in the first quadrant bounded by the x-axis, the line x = ln(π), and the curve y = sin(ex) is rotated about the x-axis. What is the volume of the generated solid?

Answer 1

Have you considered setting up the integral?

Answer 2

yes I did

Answer 3

What did you set up?

Answer 4

v = ∫[0,lnπ] π sin^2(e^x) dx

Answer 5

not sure if thats correct

Answer 6

Are you sure it's finite?

Answer 7

I think so

Answer 8

its the volume bounded by two graphs so im pretty sure

Answer 9

Good. Do you believe there is a nice, closed-form expression for the integral?

Answer 10

yes

Answer 11

do you think the integral I set up is correct?

Answer 12

Your integral looks fine.
Are you sure this isn't an integral for numerical methods?

Answer 13

Not sure

Answer 14

Okay, one more question. Is it a Definite Integral? In other words, does it actually exist at x = 0 or is that a limit behavior we need to worry about?

Answer 15

it is a definite integral

Answer 16

Well, I believe we are out of luck on this one. With a little transformation (t = e^x), we can make this one look like sin(t)/t, and that's not encouraging.

It's time for numerical methods. What tools have you?

Answer 17

??

Answer 18

There is a lot of information on the "Sine Integral".

http://mathworld.wolfram.com/SineIntegral.html

No easy form as a result.