曲线y=x^2与直线y=2x围成的平面图形饶y轴旋转一周所得旋转体的体积V

Question

曲线y=x^2与直线y=2x围成的平面图形饶y轴旋转一周所得旋转体的体积V

Answer 1

y = x² = 2x
x² - 2x = x(x - 2) = 0, x = 0, x = 2
交点O(0, 0), A(2, 4)
绕y轴旋转, 用y做自变量较为方便
在y处(0 < y < 4), 旋转体的截面为外径R = x=√y, 内径r = x = y/2 的圆环
截面积 = π(R² - r²) =π(y - y²/4)
旋转体的体积 V = ∫₀⁴π(y - y²/4)dy
= π(y²/2 - y³/12)|₀⁴
= 8π/3