※ 引述《yakiniku (燒肉哥)》之銘言：
: 1.∫[0,1]1/x^p dx (p>0)
如果0<p<1
x^(-p+1) 1 1
- ─────| = - ─
-p+1 0 1-p
1
如果p > = 1 (p=1 : 積分 = lnx| 發散)
發散 0
: 2.x^2+(y-b)^2=R^2 (0<R<b) 繞X軸旋轉之體積
torus
分成 x_右 = √[R^2 - (y-b)^2]
和 x_左 = -√[R^2 - (y-b)^2]
體積
= ∫(x_右 - x_左)2πydy
b+R
= 4π∫√[R^2 - (y-b)^2]ydy
b-R
= 4π[ ∫√[R^2 - (y-b)^2] (y-b)d(y-b) + b∫√[R^2 - (y-b)^2]dy ]
b+R 令u = (y-b)/R
= 4π[ (-2/3)√[R^2 - (y-b)^2]| + bR^2∫√[1 - u^2] du ]
b-R
令u = sint
π/2
= 4πbR^2 ∫ (1/2)(1+cos(2t))dt
-π/2
π/2
= 2πbR^2 [t + (1/2)sin(2t)]|
-π/2
= 2π^2 b R^2
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