Answer to Question #119363 in Calculus for Olivia

Question

The answer to ∬D ysin(xy)dA where D={(x,y)∈R^2:1≤x≤2,0≤y≤π}

is
Select one:
a. 0
b. (3π)/2


c. π/2

d. π

e. 1

Answer 1

$\begin{aligned} &\iint y \sin (xy)dA\textcolor{blue}{=\int_{0}^{\pi}\textcolor{red}{ \int_{1}^{2} y \sin (xy) dx}dy }\\[1 em] \textcolor{red}{ \int_{1}^{2} y \sin (xy)dx}& =\frac{-y}{y}\cos (xy)\mid _{1}^{2} =-\cos (xy)\mid _{1}^{2}\\[1 em] &=-(\cos 2y -\cos y) =-\cos 2y +\cos y\\[1 em] \textcolor{blue}{ \int_{0}^{\pi} -\cos 2y +\cos y dy} &=\frac{-1}{2}\sin 2y+\sin y\mid _{0}^{\pi}\\[1 em] &=(\frac{-1}{2}\sin 2\pi+\sin \pi)-(\frac{-1}{2}\sin 0+\sin 0)=\fcolorbox{red}{aqua}{0}\\[1 em] \text{The answer is }\fcolorbox{red}{aqua}{a} \end{aligned}$