設弓形的底為 b, 高為 h 。有個弓形面積的近似計算公式是
S = 2bh/3 + h^3/(2b)
這個公式是怎麼推導出來的呢？

解：

假設 h << b

圓的半徑 R = (b^2+4h^2)/(8h)                      (1)
R-h = (b^2-4h^2)/(8h)                             (2)

弓形方程 y = R(1-(x/R)^2)^0.5 - (b^2-4h^2)/(8h)   (3)

弓形面積 S = 2Int(0,b/2)[ydx]
= 2RInt(0,b/2)[(1-(x/R)^2)^0.5dx] - (b^3-4bh^2)/(8h)
= 2RInt(0,b/2)[1-(x/R)^2/2 - (x/R)^4/8]dx - (b^3-4bh^2)/(8h)
= bR -(b/2)^3/(3R) - (b/2)^5/(20R^3) - (b^3-4bh^2)/(8h)
= (b^3+4bh^2)/(8h) -(b/2)^3/(3R) - (b/2)^5/(20R^3) - (b^3-4bh^2)/(8h)
= bh - hb^3/(3(b^2+4h^2)) - b^5/(20R^3 2^5)
= bh - hb/(3(1+4h^2/b^2)) - b^5/(20R^3 2^5)

= bh - (hb/3)(1-4h^2/b^2+16h^4/b^4) - b^5/(20R^3 2^5)
= 2bh/3 + 4h^3/(3b)-16h^5/(3b^3) - 4h^3/(5b(1+4h^2/b^2))
= 2bh/3 + 4h^3/(3b)-16h^5/(3b^3) - 4h^3/(5b)
= 2bh/3 + 8h^3/(15b)-16h^5/(3b^3)

S = 2bh/3 + 8h^3/(15b)-16h^5/(3b^3)               (4)

如果忽略高次項，

S = 2bh/3 + 8h^3/(15b)                            (5)

為了方便記憶，考慮到高次項是負的，把第二項減小成
          8h^3/(16b)
這樣，我們就得到

S = 2bh/3 + h^3/(2b)