如何用定积分求球体的表面积和体积?

2024-12-04 16:41:13
推荐回答(2个)
回答1:

圆的方程x^2+y^2=r^2,所以y=f(x)=(r^2-x^2)^(1/2)
S=2∫(0,r)2πf(x)[1+(f'(x))^2]^(1/2)dx
=4π∫(r^2-x^2)^(1/2)*[1+x^2/(r^2-x^2)]^(1/2)dx
=4π∫(r^2-x^2)^(1/2)*[r^2/(r^2-x^2)]^(1/2)dx
=4π∫(0,r)rdx
=4πr^2

V=2∫(0,r)πf(x)^2dx=2∫(0,r) π(r^2-x^2)dx=(4/3)πr^3

回答2:

用球积分。
S=∫∫R^2sin(theta)d(theta)d(phi)
V=∫∫∫r^2sin(theta)drd(theta)d(phi)
theta从0到pi,phi从0到2pi,r从0到R