The volume of the solid surrounded by the surface  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        ---(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  ---(4)

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

Volume        ---(5)

Now apply cartesion to spherical conversion in equation (5)

       ---(6)

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

Volume

⇒ Volume

⇒ Volume

On solving above integrals we get.

⇒ Volume  unit cube