Section 15.2 : Iterated Integrals
8. Compute the following double integral over the indicated rectangle.
\[\iint\limits_{R}{{xy\cos \left( y \right) - {x^2}\,dA}}\hspace{0.5in}R = \left[ {1,2} \right] \times \left[ {\frac{\pi }{2},\pi } \right]\]Show All Steps Hide All Steps
Start SolutionThe order of integration was not specified in the problem statement so we get to choose the order of integration. As we discussed in the first few problems of this section this can be daunting task and in those problems the order really did not matter. The order chosen for those problems was mostly a cosmetic choice in the sense that both orders were had pretty much the same level of difficulty.
With this problem however, we have a real difference in the orders. If we do the \(y\) integration first we will have to do integration by parts. On the other hand, the \(x\) integration is a very simple Calculus I integration.
So, it looks like integrating with respect to \(x\) first is the way to go. Here is the integral set up to do the \(x\) integration first.
\[\iint\limits_{R}{{xy\cos \left( y \right) - {x^2}\,dA}} = \int_{{\frac{\pi }{2}}}^{\pi }{{\int_{1}^{2}{{xy\cos \left( y \right) - {x^2}\,dx}}\,dy}}\] Show Step 2Okay, let’s do the \(x\) integration now.
\[\begin{align*}\iint\limits_{R}{{xy\cos \left( y \right) - {x^2}\,dA}} & = \int_{{\frac{\pi }{2}}}^{\pi }{{\int_{1}^{2}{{xy\cos \left( y \right) - {x^2}\,dx}}\,dy}}\\ & = \int_{{\frac{\pi }{2}}}^{\pi }{{\left. {\left( {\frac{1}{2}{x^2}y\cos \left( y \right) - \frac{1}{3}{x^3}} \right)} \right|_1^2\,dy}} = \int_{{\frac{\pi }{2}}}^{\pi }{{\frac{3}{2}y\cos \left( y \right) - \frac{7}{3}\,dy}}\end{align*}\]Note that for this example, unlike the previous one, the integration by parts did not go away after doing the first integration. Be careful to not just expect things like integration by parts to just disappear after doing the first integration. They often won’t, and, in fact, it is possible that they might actually show up after doing the first integration!
Show Step 3Now all we need to take care of is the \(y\) integration. As noted this is integration by parts for the first term and so we should probably split up the integral before doing the integration by parts.
Here is the work for this problem.
\[\begin{align*}\iint\limits_{R}{{xy\cos \left( y \right) - {x^2}\,dA}} & = \int_{{\frac{\pi }{2}}}^{\pi }{{\frac{3}{2}y\cos \left( y \right)\,dy}} - \int_{{\frac{\pi }{2}}}^{\pi }{{\frac{7}{3}\,dy}}\hspace{0.25in}u = \frac{3}{2}y\,\,\,\,\,\,\,\,dv = \cos \left( y \right)dy\\ & = \left. {\left( {\frac{3}{2}y\sin \left( y \right) + \frac{3}{2}\cos \left( y \right) - \frac{7}{3}y} \right)} \right|_{\frac{\pi }{2}}^\pi \\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{3}{2} - \frac{{23}}{{12}}\pi = - 7.5214}}\end{align*}\]Note that we gave the \(u\) and \(dv\) for the integration by parts work but are leaving the details to you to verify the result.