Define $f\left(x\,y\,z\right)$ as the integrand

$f\left(x\,y\,z\right)\=1\+2 x^2\+3 y^2\+4 z^2$$\stackrel{\text{assign as function}}{\to }$$f$

Form a Riemann sum

$q\=\sum _{i\=1}^u\sum _{j\=1}^v\sum _{k\=1}^{w}f\left(1\+\frac{i}{u}\,-1\+\frac{4 j}{v}\,2\+\frac{2 k}{w}\right)\cdot \left(\frac{1\cdot 4\cdot 2}{u v w}\right)$$\stackrel{\text{assign}}{\to }$

Obtain an iterated limit

$\mathrm{limit}\left(q\,\left\{u\=\infty \,v\=\infty \,w\=\infty \right\}\right)$ = $400$$$

$\mathrm{simplify}\left(q\right)$ = $\frac{8}{3}\frac{150u^2{v}^2{w}^2\+72u^2{v}^{2}w\+36u^{2}v{w}^2\+9u{v}^2{w}^2\+8u^2{v}^2\+24u^2{w}^2\+v^2{w}^2}{u^2{v}^2{w}^2}$$$