y=x的平方与y=9-x的平方绕y轴旋转体体积公式

Question

y =x 的平方与y =9-x 的平方绕y 轴旋转体体积公式

Answer 1

y = x² = 9 - x², 2x² = 9, x = ±3/√2
二者交于A(3/√2, 9/2), B(-3/√2, 9/2)
绕y 轴旋转, 用y做自变量较方便
y = x², x = √y
y = 9 - x², x = √(9 - y)
y = 9/2上下部分的y处, 旋转体截面分别为以√(9 - y), √y为半径的圆
V = V1 + V2
= ∫π(√y)²dy + ∫π[√(9 -y)]²dy (区间分别为0 -> 9/2, 9/2 ->9)
= πy²/2 (0 -> 9/2) + π(9y - y²/2) (9/2 ->9)
= 81π/8 + 81π/8
=81π/4