Volume of solid rotated around y=1

[jaguar ride]

Homework Statement

Find the volume of the solid formed by revolving the region bounded by f(x) = 2-x^2 and g(x) = 1 about the line y = 1.

Homework Equations

V = ∏∫(1-f(x))^2dx - ∏∫(1-g(x))^2dx

The Attempt at a Solution

I keep ending up with ∏∫(1-(2-x^2))^2dx - ∏∫(1-1)^2dx, on the interval from x = 0, x = 1. This gives me ∏∫[(-1+x^2)^2 - 0]dx
= ∏[x - (2/3)x^3 + (1/5)x^5].

This keeps giving me 8∏/15.
This is on a test review sheet, with the answer being 16∏/15. So I can get that answer by multiplying mine by 2, but why would I do that? I must be doing something wrong somewhere else. Any help would be greatly appreciated! Thanks.

 

[Saitama]

Hi jaguar ride! Welcome to PF!

but why would I do that? I must be doing something wrong somewhere else.

Sketch a plot of the given region. ;)

You will notice that the integration limits are incorrect.

 

[jaguar ride]

Ah yes. Should have mentioned that a sketch is the first thing I do when solving these problems.
That being said, I feel extra stupid for staring at this thing for so long and not realizing the limits could be negative.

Got the answer. Makes sense that I could multiply it by 2 now...

Thanks for your help and the warm welcome, I'll be back for physics help soon enough...

 

[Saitama]

Ah yes. Should have mentioned that a sketch is the first thing I do when solving these problems.
That being said, I feel extra stupid for staring at this thing for so long and not realizing the limits could be negative.

Got the answer. Makes sense that I could multiply it by 2 now...

Thanks for your help and the warm welcome, I'll be back for physics help soon enough...

Glad to help!