Help with washer method

[R.M.]

I am not certain that I have setup this problem correctly.

Problem: "Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
y=e^-x, y=1, x=2, about y =2.

I sketched the region, sketched the resulting solid, and drew a typical washer (shown below).


I have R = 2, r = 2-e^-x, and I am integrating between 0 and 2. The area is [MATH]\pi[/MATH](R²-r²) = [MATH]\pi[/MATH](4e^-x+e^-2x).
V = [MATH]\pi\int_0^2 (4e^{-x}+e^{-2x})\, dx [/MATH]
I can take it from here, but I'm not quite certain that my integral is setup correctly.

 

[Dr.Peterson]

Looks right.

Evidently you did the sketches you didn't show, correctly. (Showing them, and your work simplifying the expression, might have saved me a little time.)

Now go for it!

 

[R.M.]

Knew I forgot something.

 

[Dr.Peterson]

Yes, those are what I expected to see. Now just do the integration, which shouldn't be hard.