\(\text{We have to evaluate the integral }\)

\(\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx\)

\(\int_1^3 x(\int_0^{\pi/2}\sin y\ dy)dx\)

\(\int_1^3 x[-\cos y]_0^{\pi/2}dx\)

\(-\int_1^3 x(0-1)dx\)

\(\int_1^3 xdx\)

\([\frac{x^2}{2}]_1^3\)

\(=\frac{9}{2}-\frac{1}{2}\)

\(=\frac{8}{2}\)

\(=4\)

\(\text{therefore}\)

\(\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx=4\)

\(\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx\)

\(\int_1^3 x(\int_0^{\pi/2}\sin y\ dy)dx\)

\(\int_1^3 x[-\cos y]_0^{\pi/2}dx\)

\(-\int_1^3 x(0-1)dx\)

\(\int_1^3 xdx\)

\([\frac{x^2}{2}]_1^3\)

\(=\frac{9}{2}-\frac{1}{2}\)

\(=\frac{8}{2}\)

\(=4\)

\(\text{therefore}\)

\(\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx=4\)