Question

Calc 3 question, please help! Iterated Integrals

Answer 1

\[\int\limits_{0}^{\pi}\int\limits_{1}^{2}ysin(xy)dxdy\]

Answer 2

I need help integrating with respect to x

Answer 3

\[\begin{align*}\int\limits_{0}^{\pi}\int\limits_{1}^{2}y\sin(xy)~dx~dy&=\int\limits_{0}^{\pi}\int\limits_{y}^{2y}y\sin u~\left(\frac{du}{y}\right)~dy&\text{set }u=xy\\\\
&=\int\limits_{0}^{\pi}\int\limits_{y}^{2y}\sin u~du~dy\\\\
&=-\int\limits_{0}^{\pi}[\cos u]_y^{2y}~dy\\\\
&=-\int\limits_{0}^{\pi}(\cos2y-\cos y)~dy\\\\
&=\int\limits_{0}^{\pi}(\cos y-\cos2y)~dy
\end{align*}\]