The volume of the solid surrounded by the surface \({\left( {\frac{x}{a}} \right)^{2/3}} + {\left( {\frac{y}{b}} \right)^{\frac{2}{3}}} + {\left( {\frac{z}{c}} \right)^{2/3}} = 1\) is

Concept:

Cartesian system (a, b, c) to spherical coordinate system (r, θ, ϕ) conversion.

a = r sin θ cos ϕ, b = r sin θ sin ϕ

c = r cos θ [note: a, b, c are generalised cartesian parameter different then given solid parameter]

da ⋅ db ⋅ dc = r²sin θ dr ⋅ dθ ⋅ dϕ

Calculation:

Given solid is \({\left( {\frac{x}{a}} \right)^{\frac{2}{3}}} + {\left( {\frac{y}{b}} \right)^{\frac{2}{3}}} + {\left( {\frac{z}{c}} \right)^{\frac{2}{3}}} = 1\)       ---(1)

Now, put x = au³, y = bv³ & z = cω³       ---(2)

⇒ dx = 3a4²⋅d4, dy = 3bv²⋅dv & dz = 3cw²dw       ---(3)

Put equation (2) in equation (1), then we get

u² + v² + w² = 1 (which is equation of sphere)

Since volume enclosed by solid can be calculate as,

Volume \(=\iiint{dx\cdot dy\cdot dz}\) ---(4)

So put equation (3) in equation (4), then we get

Volume \(=\iiint{\left( 27 \right)\left( abc \right)\left( {{u}^{2}}{{v}^{2}}{{w}^{2}} \right)du\cdot dv\cdot dw}\)       ---(5)

Now apply cartesion to spherical conversion in equation (5)

\(\left. \begin{matrix} u=r\sin \theta \cos \phi ,~v=r\sin \theta \sin \phi ~\And ~w=r\cos \theta \\ du\cdot dv\cdot dw={{r}^{2}}\cdot \sin \theta \cdot dr\cdot d\theta \cdot d\phi \\ \end{matrix} \right\}\)       ---(6)

Put equation (6) into equation (5) then we get

Volume \(=\left( 27~abc \right)\underset{r}{\overset{{}}{\mathop \int }}\,\underset{\theta }{\overset{{}}{\mathop \int }}\,\underset{\phi }{\overset{{}}{\mathop \int }}\,\left( {{r}^{2}}\cdot {{\sin }^{2}}\theta \cdot {{\cos }^{2}}\phi \right)\cdot ({{r}^{2}}{{\sin }^{2}}\theta {{\sin }^{2}}\phi )\cdot ({{r}^{2}}{{\cos }^{2}}\theta ){{r}^{2}}\sin \theta drd\theta d\phi\)

⇒ Volume \(\left( 27~abc \right)~\mathop{\int }_{r}^{{}}\mathop{\int }_{\theta }^{{}}\mathop{\int }_{\phi }^{{}}\left( {{r}^{8}} \right)\cdot \left( {{\sin }^{2}}\phi \cdot {{\cos }^{2}}\phi \right)\cdot \left( {{\sin }^{5}}\theta \cdot {{\cos }^{2}}\theta dr\cdot d\theta ~d\phi \right)\)

⇒ Volume \(=\left( 27~abc \right)\underset{r=0}{\overset{1}{\mathop \int }}\,{{r}^{8}}\cdot dr\underset{\theta =0}{\overset{\pi /2}{\mathop \int }}\,{{\sin }^{5}}\theta \cdot {{\cos }^{2}}\theta d\theta ~\underset{\phi =0}{\overset{\frac{\pi }{2}}{\mathop \int }}\,{{\sin }^{2}}\phi \cdot {{\cos }^{2}}\phi d\phi\)

On solving above integrals we get.