Question

一道高数题

定积分的应用，求旋转体体积，要具体过程。

Answer 1

代吉米多维奇《数学分析习题集》第2482题的公式
V = (2π/3)∫<α, β>r^3(θ)sinθdθ
得 V = (2π/3)∫<0, π/2>64(1+cosθ)^3 sinθdθ
= -(2π/3)∫<0, π/2>64(1+cosθ)^3d(1+cosθ)
= -(32π/3)[(1+cosθ)^4]<0, π/2> = 160π

Answer 2

教材上或者习题中应该有极坐标下的旋转体的体积计算公式（或可自己推导），依样画葫芦就是。

Answer 3

欧拉

Answer 4

(x^2+1)/[(x^2-1)(x+1)]
=1/(x+1) + 2/[(x^2-1)(x+1)]
let
2/[(x^2-1)(x+1)]≡ A/(x+1) +B/(x+1)^2 + C/(x-1)
=>
2 ≡ A(x+1)(x-1) +B(x-1) + C(x+1)^2
x=1, C=1/2
x=-1, B=-1
coef. of x^2
A+C =0
A= -1/2
2/[(x^2-1)(x+1)]≡ -(1/2)[1/(x+1)] -1/(x+1)^2 + (1/2)[1/(x-1)]
(x^2+1)/[(x^2-1)(x+1)] ≡ (1/2)[1/(x+1)] -1/(x+1)^2 + (1/2)[1/(x-1)]
∫(x^2+1)/[(x^2-1)(x+1)] dx
=∫ { (1/2)[1/(x+1)] -1/(x+1)^2 + (1/2)[1/(x-1)] } dx
=(1/2)ln|x^2-1| +1/(x+1) + C