先把y=x, y=x^2, x=2三条曲线的交点求到,总共有四个交点:
(0,0), (1,1), (2,2),(2,4),以x作为积分变量,每个积分微元dx对应的是一个圆环(薄片),
其体积为:
dV = | π (x+1)^2 - π(x^2+1)^2 | dx
V = ∫(0, 2) | π (x+1)^2 - π(x^2+1)^2 | dx
= ∫ (0, 1) π[ (x+1)^2 - (x^2+1)^2] dx + ∫ (1, 2) π[ (x^2+1)^2-(x+1)^2 ] dx
= π( x^2 -x^5/5-x^3/3) | (0,1) + π ( x^5/5+x^3/3-x^2 ) | (1,2)
= 7π/15 + 83π/15
=6π