\(\displaystyle\int\limits^1_{y=0} \) \(\displaystyle\int\limits^1_{\mathrm x=y^2}\) \(\displaystyle\int\limits^{1-\mathrm x}_{z=0}\) x dz dx dy
\(=\displaystyle\int\limits^1_0\left(\int\limits^1_{y^2}\mathrm x \,\left[z\right]^{1-\mathrm x}_0d\mathrm x\right)dy\)
\(=\displaystyle\int\limits^1_0\left(\int\limits^1_{y^2}\mathrm x(1-\mathrm x)d\mathrm x\right)dy\)
\(=\displaystyle\int\limits^1_0\left(\frac{\mathrm x^2}{2}-\frac{\mathrm x^3}{3}\right)^1_{y^2}dy\)
\(=\displaystyle\int\limits^1_0\left(\frac{1}{2}(1-y^4)-\frac{1}{3}(1-y^6)\right)dy\)
\(=\frac{1}{2}\displaystyle\int\limits^1_0(1-y^4)dy\) \(-\frac{1}{3}\displaystyle\int\limits^1_0(1-y^6)dy\)
\(=\frac{1}{2}\left[y-\frac{y^5}{5}\right]^1_0\) \(-\frac{1}{3}\left[y-\frac{y^7}{7}\right]^1_0\)
\(=\frac{1}{2}\left(1-\frac{1}{5}\right)-\frac{1}{3}\left(1-\frac{1}{7}\right)\)
\(=\frac{1}{2}\times \frac{4}{5}-\frac{1}{3}\times \frac{6}{7}\)
\(=\frac{2}{5}-\frac{2}{7}\) \(=\frac{14-10}{35}=\frac{4}{35}\)