定积分在几何上的应用题：求曲线y=sin^x(0《x《兀)与直线y=0所围成的图形分别绕x轴、y轴旋转而成的体积

Question

如题所述

Answer 1

绕x轴旋转：V=∫(0,π) π(sinx)^2 dx
=π/2*∫(0,π) (1-cos2x) dx
=π/2*(x-sin2x/2)|(0,π)
=π/2*(π)
=(π^2)/2
绕y轴旋转：V=∫(0,1) π(f(y))^2 dy - ∫(0,1) π(g(y))^2 dy
=π*[∫(π,π/2) x^2*cosxdx - ∫(0,π/2) x^2*cosxdx]
=π*{[x^2sinx + 2(xcosx-sinx)]|(π,π/2)-[x^2sinx + 2(xcosx-sinx)]|(0,π/2)}
=π*[(π^2/4-2+2π)-(π^2/4-2)]
=2π^2