Evaluate the double integral ${\displaystyle \underset{R}{\overset{\phantom{}}{}}}ycosxdA$ of the function over the rectangle $R=[0,\frac{}{4}][0,1].$

Answer: $\frac{\sqrt[\phantom{2}]{2}}{4}$

$3.$ Evaluate the double integral ${\displaystyle \underset{R}{\overset{\phantom{}}{}}}{e}^{7x+2y}dA$ of the function over the rectangle $R=[0,5]x[0,2].$ Answer: $\frac{1}{14}(e^{35}-1)(e^4-1)$

$4.$ Evaluate the double integral ${\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}{\displaystyle \underset{0}{\overset{cosx}{}}}(1+sinx)dydx.$

Answer: $\frac{3}{2}$