|x| < 1: y = 3 - (1 - x²) = x² + 2
|x| ≥ 1: y = 3 - (x² - 1) = 4 - x²
y = 0, x = ±2
与x轴交于A(2, 0), B(-2, 0)
y = 3, x = ±1,曲线与y = 3交于C(1, 3), D(-1, 3)
曲线与x轴围成的封闭图形在A, B, C, D之间
显然旋转体关于y轴对称, 这里只考虑0 ≤ x ≤2,结果加倍即可。
(a) 取x, 0 ≤ x ≤1
旋转体的截面是以3为外径(y= 3与x轴的距离), 以1- x²为内径(y = 3与y = x² + 2的距离)的圆环
截面积S = π[3² - (1 - x²)²]
v1 = ∫₀¹Sdx = ∫₀¹π[3² - (1 - x²)²]dx
= π∫₀¹(8 + 2x² - x⁴)dx
= π(8x + 2x³/3 - x⁵/5)|₀¹
= 127π/15
(b) 取x, 1 ≤ x ≤2
旋转体的截面是以3为外径(y= 3与x轴的距离), 以x² - 1为内径(y = 3与y = 4 - x²的距离)的圆环
截面积S = π[3² - (1 - x²)²]
v2 = ∫₁²Sdx = ∫₁²π[3² - (x² - 1)²]dx
= π∫₁²(8 + 2x² - x⁴)dx
= π(8x + 2x³/3 - x⁵/5)|₁²
= 97π/15
V= 2(v1 + v2)
= 448π/15