Finding Volume of Rotated Solid: Disk Method

[xibalba1]

Homework Statement

Consider the solid obtained by rotating the region bounded by the given curves about y=10

Find the volume V of this solid using DISK METHOD

Homework Equations

y=5e^-x, y=5, x=4

The Attempt at a Solution

setting up the picture easy. but I'm having trouble knowing HOW to find the radius. any help would be cool. thank you.

 

[lanedance]

draw a horizontal line from y=10, then draw a vertical line down & see where it intersects the region, the radius is the vertical disatnce from y=10 to the point in your region

from that you should find an innner (ri) & outer (r0) radius, that may be both be functions of x.

then set up the integral over x

 

[lanedance]

corrected above after drawing it

 

[xibalba1]

Okay, I think I know what you are saying. so far I got pitimes the integral from 0 to 4 of [(10-5e^(-x)]^2 -(5)^2 dx

what do you think

 

[Raskolnikov]

Okay, I think I know what you are saying. so far I got pitimes the integral from 0 to 4 of [(10-5e^(-x)]^2 -(5)^2 dx

what do you think

Yep, that's it!

 

[xibalba1]

Dude man, I keep getting the wrong answer, or at least www.webassign.net tells me so. SO...the problem must lie within my algebra.

I freaking hate webassign.net. You do all this work by setting up the problem, doing the calculus, etc, but it aint worth crap because you get the final answer incorrect. B.S. man, you know what I mean?

By the way, www.webassign.net is my MANDATORY online homework I have to do. No TEXT BOOK problems were assigned this semester. They're all on webassign. Argh.

 

[Raskolnikov]

$$V = \pi \int_0^4 \left[ (100 - 100e^{-x} + 25e^{-2x}) - 25 \right] dx$$

$$= \pi \int_0^4 25e^{-x}(e^{-x} - 4)dx \ + \ 75\pi x|_0^4$$

Let $$u = e^{-x} - 4 \ \Rightarrow \ du = -e^{-x}dx.$$

So $$V = 300\pi - 25 \pi \int u du$$

$$= 300\pi - 25 \pi \left( \frac{(e^{-x} - 4)^2}{2} \right) \right|_0^4.$$

Does that help?