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
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