Question

高等数学问题

第四题旋转体问题，请给出详细步骤谢谢

Answer 1

y = -x^2+2x 交 x 轴于 O(0, 0), A(2, 0)
y = 1 - (x-1)^2, 抛物线顶点 P(1, 1), 即 x = 1 ± √(1-y)。
Vx = π∫<0, 2>(-x^2+2x)^2dx = π∫<0, 2>(x^4-4x^3+4x^2)dx
= π[x^5/5-x^4+4x^3/3]<0, 2> = 16π/15.
Vy = π∫<0, 1>{[1+√(1-y)]^2 - [1-√(1-y)]^2}dy
= 4π∫<0, 1>√(1-y)dy [令 √(1-y) = u, 则 y = 1-u^2]
= 4π∫<1, 0>(-2u^2)du = 8π∫<0, 1>u^2du = 8π/3.
或 Vy = ∫<0, 2>2πx(-x^2+2x)dx = 2π∫<0, 2>(-x^3+2x^2)dx
= 2π[-x^4/4+2x^3/3]<0, 2> = 8π/3.