Find the volume of the solid obtained by rotating the region bounded by the curves y=cos(x), y=0, x=0, and x=π/2 about the line y=1.

Answer 1

radius of cylinder of skin thickness dr (r goes from 0 to 1)
r = 1-cos x
dr = sin x dx
so circumference = 2 π (1-cos x)
height of that cylinderical shell = x
dV = 2 π(1-cos x) x dr
= 2 π (1-cos x)x sin x dx
= 2 π x sin x dx - 2 π x sin x cos x dx

= 2 π [ sin x-x cos x]
- 2 π/8 [ sin 2 x - 2 x cos 2x]
(I used wolfram alpha integral - google it)
evaluate at x = pi/2
= 2 π [ 1]
- 2 π/8 [ 0 + π ]

= 2 π - π^2 /4
now
evaluate at x = 0
= 2 π[ 0 ]
- π/4 [ 0 ] handy
so
2 π - π^2 /4

Answer 2

lots of algebra here
here is the integral site:
http://www.wolframalpha.com/calculators/integral-calculator/

Answer 3

works for me. I did the integral directly, using

∫[0,1] 2π(1-y) arccos(y) dy

and got the same answer.

I like your technique of avoiding the arccos, though.