x^2+Y^2=9,求此圆绕x=-4一周的旋转体体积,用定积分求

题目

x^2+Y^2=9,求此圆绕x=-4一周的旋转体体积,用定积分求

答案

By Disc Method：
x²+y²=9 => x²=9-y² => x=±√(9-y²)
V=π∫(-3,3) {[√(9-y²)+4]²-[-√(9-y²)+4]²} dy
= 16π∫(-3,3) √(9-y²) dy
= 16π * 9π/2
= 72π²
By Shell Method：
x²+y²=9 => y=±√(9-x²),Shell radius = x-(-4) = x+4
V=2π∫(-3,3) (x+4)[√(9-x²)-(-√(9-x²))] dx
= 2π∫(-3,3) [2x√(9-x²)+8√(9-x²)] dx
= 2π * 36π
= 72π²