Question

求圆x平方＋（5-y）平方=16绕x轴旋转一周所产生的旋转体的体积

Answer 1

y=5±√(16-x^2)

V=π∫(-1,1)((5+√(16-x^2))-(5-√(16-x^2)))dx

V=2π∫(-1,1)√(16-x^2)dx

V=4π∫(0,1)√(16-x^2)dx

先抛开积分x的上下限

设x=4sint,t属于(-π/2,π/2)

sint=x/4

cost=√(16-x^2)/4

V=4π∫4costd(4sint)

V=64π∫(cost)^2dt

V=32π∫[cos(2t)+1]dt

V=16π∫cos(2t)d(2t)+32π∫1dt

V=16πsin(2t)+32πt

V=32π(sintcost+t)

V=32π[x√(16-x^2)/16+arcsin(x/4)]

V=2πx√(16-x^2)+32πarcsin(x/4)

代入积分上下限(0,1)

得V=2π√15+32πarcsin(1/4)