Question

Evaluate `int int int sqrt(1-x^2/a^2-y^2/b^2-x^2/c^2 )`dx dy dz over the ellipsoid `x^2/a^2+y^2/b^2+z^2/c^2=1.`

Answer 1

Ellipsoid : ^{`x^2/a^2+y^2/b^2+z^2/c^2`}

Cartesian coordinates → spherical coordinate system
                         (𝒙,𝒚,𝒛) → (𝒓,𝜽,∅)

Put 𝒙=𝒂 𝒓𝒔𝒊𝒏 𝜽 𝒄𝒐𝒔 ∅ ,𝒚=𝒃 𝒓 𝒔𝒊𝒏 𝜽 𝒔𝒊𝒏 ∅ ,𝒛=𝒄 𝒓𝒄𝒐𝒔 𝜽

𝒅𝒙 𝒅𝒚 𝒅𝒛=𝒂𝒃𝒄 𝒓^𝟐 𝒔𝒊𝒏 𝜽 𝒅𝒓 𝒅𝜽 𝒅∅

`therefore x^2/a^2+y^2/b^2+z^2/c^2=r^2`

`f(x,y,z)=sqrt(1-x^2/a^2-y^2/b^2-x^2/c^2 )=sqrt(1-r^2)=f(r,theta,O/`

Limits :
𝟎 ≤ 𝒓 ≤ 𝟏
𝟎 ≤ 𝜽 ≤ `pi/2`
𝟎 ≤ ∅ ≤ `pi/2`

`therefore  "I" = 8 int int int sqrt(1-r^2) abc  r^2 sin theta drd thetad O/`

`= 8 int_0^(pi/2) int_0^(pi/2) int_0^r sqrt(1-r^2) abc  r^2 sin theta drd thetad O/`

`=8abcint_0^(pi/2)sinthetad thetaint_0^(pi/2) dO/int_0^rsqrt(1-r^2)r^2 dr`

`8abc[-costheta]_0^(pi/2) [O/]_0^(pi/2)int_0^(pi/2)costsin^2tcost dt` ---------{ put r = sint}

`=8abc(pi/2)(pi/8)` …………{`usibetaf^n`}

`therefore"I"=pi^2/2(abc)`