切线与x轴的交点为B(-1, 0), 与y轴的交点为C(0, 1/2)
切线(y = (x+1)/2)与抛物线和坐标轴所围的区域分别为绿色和紫色.
绕x轴旋转一周,在x处截面积为:
f(x) = π[(x+1)/2]² = π[(x+1)²/4 (-1 ≤ x < 0)
= π[(x+1)/2]² - π(√x)² = π[(x+1)²/4 - πx = π[(x-1)²/4 (0 ≤ x ≤ 1)
V = ∫ π(x+1)²dx/4 (从-1到0)
+ ∫ π[(x-1)²dx/4 (从0到1)
= (π/4)∫ (x+1)²d(x+1) (从-1到0)
+ (π/4)∫ (x-1)²d(x-1) (从0到1)
= π(x+1)³/12 (从-1到0)
+ π(x-1)³/12 (从0到1)
= π/12 + π/12
= π/6
