Question

Find the following integral

Answer 1

There's nothing preventing us from computing one integral at a time:

`\displaystyle \int_0^(2-x) xyz \,\mathrm dz = \frac12xyz^2\bigg|_(z=0)^(z=2-x) \\\\ = \frac12xy(2-x)^2`

`\displaystyle \int_0^(1-x)\int_0^(2-x)xyz\,\mathrm dz\,\mathrm dy = \frac12\int_0^(1-x)xy(2-x)^2\,\mathrm dy \\\\ = \frac14xy^2(2-x)^2\bigg|_(y=0)^(y=1-x) \\\\= \frac14x(1-x)^2(2-x)^2`

`\displaystyle\int_0^1\int_0^(1-x)\int_0^(2-x)xyz\,\mathrm dz\,\mathrm dy\,\mathrm dx = \frac14\int_0^1x(1-x)^2(2-x)^2\,\mathrm dx`

Expand the integrand completely:

`x(1-x)^2(2-x)^2 = x^5-6x^4+13x^3-12x^2+4x`

Then

`\displaystyle\frac14\int_0^1x(1-x)^2(2-x)^2\,\mathrm dx = \left(\frac16x^6-\frac65x^5+\frac{13}4x^4-4x^3+2x^2\right)\bigg|_(x=0)^(x=1) \\\\ = \boxed{(13)/(240)}`