1 Answers
$$\iiint_E 6xy,dV
= \int_0^1\int_0^{\sqrt x}\int_0^{1+x+y} 6xy,dz,dy,dx
= \int_0^1\int_0^{\sqrt x} 6xy(1+x+y),dy,dx$$
$$= 6\int_0^1 x\left[\frac{y^2}{2}(1+x)+\frac{y^3}{3}\right]_0^{\sqrt x}dx
= 6\int_0^1 x\left(\frac{x}{2}(1+x)+\frac{x^{3/2}}{3}\right)dx$$
$$= 6\int_0^1 \left(\frac{x^2}{2}+\frac{x^3}{2}+\frac{x^{5/2}}{3}\right)dx
= 6\left(\frac{1}{6}+\frac{1}{8}+\frac{2}{21}\right)
= \frac{65}{28}$$