Integration by parts

(a) ∫x cos x dx
= ∫x d(sin x)
= x sin x - ∫sin x dx
= x sin x + cos x + C
(b) By shell method,
Required capacity
= ∫2πxy dx [π,0]
= 2π∫x[1 - (- cos x)] dx [π,0]
= 2π∫(x + x cos x) dx [π,0]
= 2π∫x dx [π,0] + 2π∫x cos x dx [π,0]
= 2π[x^2 / 2][π,0] + 2π[x sin x + cos x][π,0]
= (π^3 - 0) + 2π(-1 - 1)
= π^3 - 4π


圖片參考：http://imgcld.yimg.com/8/n/HA00346210/o/701110180095813873479670.jpg


2011-10-18 23:01:56 補充：
Amendment:
Required capacity
= ∫2πx(1-y) dx [π,0]