高数,旋转体体积的定积分表达式问题 y=x^2,y=x^2+1,y=2,y轴,绕y轴旋转一周

Question

如题所述

Answer 1

y=x^2绕y轴一周的立体体积减去y=x^2+1绕y轴一周的立体体积分即可
将两曲线写为：x=√y,x=√(y-1)
dV1=π(√y)^2dy
则V1=π∫[0-->2](√y)^2dy
=π∫[0-->2]ydy
=π/2y^2 [0-->2]
=2π
dV2=π(√(y-1))^2dy
V2=π∫[1-->2](√(y-1))^2dy
=π∫[1-->2](y-1)dy
=π/2*y^2-πy [1-->2]
=π/2
因此所求体积为：V1-V2=2π-π/2=3π/2