$[12]1.$ Evaluate the integrals and sketch the region of integration.

${\displaystyle \underset{-1}{\overset{0}{}}}{\displaystyle \underset{-1}{\overset{1}{}}}(x+y+1)dxdy$

${\displaystyle \underset{0}{\overset{}{}}}{\displaystyle \underset{0}{\overset{x}{}}}xsinydydx.$

$[8]2.$ Evaluate the integral ${\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{3y}{\overset{3}{}}}{e}^{x^2}dxdy$ by reversing the order of integration.

$[12]3.$ Evaluate the iterated integral ${}_D(x^2+y^2)dA$ over the region $D$ in the first quadrant between the functions $y=2x$ and $y=x^2.$ Evaluate the iterated integral by integrating first with respect to $y$ and then integrating first with respect to $x.$

$[10]4.$ Use a double integral to find the area of $R,$ where $R$ is the region bounded by $y-$axis, $y=lnx,y=0$ and $y=1.$

$[8]5.$ Find the volume of the solid bounded above by the graph of $f(x,y)=$xysin$(x^{2}y)$ and below by the $xy-$plane on the rectangular region $D=\{(x,y)|xin[0,1],$yin$[0,]\}.$