3dt
坐標是 (x, y) 的點密度是 μ(x, y, z) = √(x^2+y^2)/(a√2).
記第壹掛限部分為 Ω1， 由對稱性
M = ∫∫∫<Ω> μ(x, y, z)dv = 8∫∫∫<Ω1> μ(x, y, z)dv
= 8∫<0, a/8>dz∫<0, a>dx∫<0, a>[√(x^2+y^2)/(a√2)]dy
= (4√2/a)∫<0, a/8>dz∫<0, a>dx∫<0, a>√(x^2+y^2)dy
= (√2/2)∫<0, a>dx∫<0, a>√(x^2+y^2)dy
引入極坐標
M = (√2/2)[∫<0, π/4>dt∫<0, a/cost>r·rdr+∫<π/4, π/2>dt∫<0, a/sint>r·rdr]
= (√2/6)a^3[∫<0, π/4>(sect)3dt + ∫<π/4, π/2>(csct)^3dt]
其中 I1 = ∫(sect)3dt = ∫sectdtant = secttant - ∫sect(tant)^2dt
= secttant - ∫sect[(sect)^2-1]dt = secttant - I1 + ∫sectdt
= secttant - I1 + ln(sect+tant)
得 I1 = (1/2)[secttant+ln(sect+tant)]
I2 = ∫(csct)3dt = -∫csctdcott = -csctcott - ∫csct(cott)^2dt
= -csctcott - ∫csct[(csct)^2-1]dt = -csctcott - I2 + ∫csctdt
= -csctcott - I2 + ln(csct-cott)
得 I2 = (1/2)[-csctcott + ln(csct-cott)]
則 M = (√2/12)a^3{[secttant+ln(sect+tant)]<0, π/4>
+ [-csctcott + ln(csct-cott)]<π/4, π/2>}
= (√2/12)a^3[√2+ln(√2+1)+√2-ln(√2-1)]
= (√2/12)a^3[2√2+2ln(√2+1)] = [1/3+(√2/6)ln(√2+1)]a^3