參考圖:http://mathworld.wolfram.com/SteinmetzSolid.html
設三圓柱半徑皆為a
由對稱性
相交部份體積即為4份全等柱體體積
由 x^2 + z^2 = a^2,得 z_1 = (a^2 - x^2)^1/2 z_2 = -(a^2 - x^2)^1/2
D = {(x,y)|x^2 + y^2 = a^2, x = y, x = -y 三者所圍出通過一四象限之扇形}
所求體積為
4∫∫ z_1 - z_2 dxdy
D
= 4∫∫ 2(a^2 - x^2)^1/2 dxdy
D
π/4 a
= 8 ∫ ∫ (a^2 - (rcosθ)^2)^1/2 rdrdθ
-π/4 0
π/4 a
= 8 ∫ -(a^2 - (rcosθ)^2)^3/2 / 3(cosθ)^2 | dθ
-π/4 0
π/4
= 8a^3 ∫ -(1 - (cosθ)^2)^3/2 / 3(cosθ)^2 + 1/ 3(cosθ)^2 dθ
-π/4
又被積式為偶函數
π/4
= 16a^3∫ -(1 - (cosθ)^2)^3/2 / 3(cosθ)^2 + 1/ 3(cosθ)^2 dθ
0
π/4
= 16a^3∫ -(sinθ)^3 / 3(cosθ)^2 + 1/ 3(cosθ)^2 dθ
0
π/4
= 16a^3∫ -(1-(cosθ)^2)sinθ/ 3(cosθ)^2 + 1/ 3(cosθ)^2 dθ
0
π/4
= (16/3)a^3 (-cosθ- secθ + tanθ)|
0
= (16/3)a^3 (-1/√2 - √2 + 1 + 1 + 1)
= 8a^3 (2 - √2)
相交部份表面積即為12片全等柱面面積
由 x^2 + z^2 = a^2,得 z = (a^2 - x^2)^1/2
其中一片柱面參數式為
→
r(x,y) = ( x , y , (a^2 - x^2)^1/2 )
on D = {(x,y)|x^2 + y^2 = a^2, x = y, x = -y 三者所圍出通過一四象限之扇形}
→ →
r_x X r_y = (x/(a^2 - x^2)^1/2 , 0 , 1)
→ →
|r_x X r_y| = a/(a^2 - x^2)^1/2
所求表面積為
12∫∫ a/(a^2 - x^2)^1/2 dxdy
D
π/4 a
= 12 ∫ ∫ ar/(a^2 - (rcosθ)^2)^1/2 drdθ
-π/4 0
π/4 a
= 12 ∫ a(a^2 - (rcosθ)^2)^1/2 / -(cosθ)^2 | dθ
-π/4 0
π/4 a^2 (1 - (cosθ)^2)^1/2 a^2
= 12 ∫ ------------------------ + ----------- dθ
-π/4 -(cosθ)^2 (cosθ)^2
又被積式為偶函數:
π/4 a^2 (1 - (cosθ)^2)^1/2 a^2
= 24 ∫ ------------------------ + ----------- dθ
0 -(cosθ)^2 (cosθ)^2
π/4 a^2 (sinθ) a^2
= 24 ∫ ------------- + ----------- dθ
0 -(cosθ)^2 (cosθ)^2
π/4
= 24(-a^2 secθ + a^2 tanθ)|
0
= 24(-a^2 √2 + a^2 + a^2 )
= 24a^2 (2 - √2)