Question

Evaluate `int int intx^2dxdydz` over the volume bounded by planes x=0,y=0, z=0 and `x/a+y/b+z/c=1`

Answer 1

Let V = `int int intx^2dxdydz`

Region of integration is volume bounded by the planes x=0, y=0, z=0 and `x/a+y/b+z/c=1`

Put x = au , y = bv , z = cw
∴ dx dy dz = abc du.dv.dw

The intersection of tetrahedron with all axes is : (1,0,0),(0,1,0),(0,0,1).
𝟎 ≤ 𝒘 ≤ (𝟏−𝒖−𝒗)
𝟎 ≤ 𝒗 ≤ (𝟏−𝒖)
𝟎 ≤ 𝒖 ≤ 𝟏
The volume required is given by ,

`"V"=int_0^1int_0^(1-u)int_0^(1-u-v)abca^2u^2du dv dw`

`=a^3bcint_0^1int_0^(1-u)(1-u-v)u^2 dv du`

`=a^3bcint_0^1u^2[v-uv-v^2/2]_0^(1-u)du`

`=a^3bcint_0^1u^2[1-u-u-u^2-(u^2(1-u)^2)/2]du`

`=a^3bc[u^3/3-u^4/2+u^5/5-1/2(u^3/3 -1/2u^4+u^5/5)]_0^1`

`therefore "V"=1/60(a^3bc)`