※ 引述《WHYGGAAYY (WHYGGAAYY)》之銘言：
: Ix =∫∫∫(y^2+z^2)dxdydz of a mass of density 1 in a region T
: T
: about the x-axis. Find the Ix when T is as follows.
: The cylinder y^2+z^2≦a^2 , 0≦x≦h
: 答案是 πha^4/2
Ix = ∫∫∫ (y^2 + z^2) dxdydz
T
2π a h
= ∫ ∫ ∫ (r^2)(r) dxdrdθ
0 0 0
(令 y = (r)(cosθ) , z = (r)(sinθ) , 則 |J| = r , 0≦r≦a , 0≦θ≦2π)
2π a h
= ∫ ∫ ∫ r^3 dxdrdθ
0 0 0
2π a |x = h
= ∫ ∫ (r^3)(x) | drdθ
0 0 |x = 0
2π a
= ∫ ∫ (h)(r^3) drdθ
0 0
2π r^4 |r = a
= ∫ (h)(-----) | dθ
0 4 |r = 0
2π a^4 a^4 |2π (a^4)(h)(π)
= ∫ (h)(-----) dθ = (h)(-----)(θ) | = ------------
0 4 4 |0 2
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