Question

高等数学 求平面图形面积，旋转体体积

Answer 1

两曲线交于 (1, 1),
D 的面积 S = ∫<0, 1>(2-x^2 - √x )dx
= [2x-(1/3)x^3-(2/3)x^(3/2)]<0, 1> = 1
D 绕 y 轴旋转体的体积
V = π∫<0, 1>y^4dy + π∫<1, 2>(2-y)dy
= π/5 + π/2 = 7π/10

Answer 2

S=∫(0，1) (2-x^2-√x)dx=∫(0，1)d(2x-x^3/3-2x√x/3)=2-1/3-2/3=1
V=∫(0,1)πx^2d√x+∫(1,2)πx^2d(2-x^2)=∫(0,1)π(√x)^4d√x-∫(1,0)2πx^3dx
=(1/5)∫(0,1)πd(√x)^5-(1/2)∫(1,0)πdx^4
=π/5+π/2=7π/10