Convenient Order of Integration

I’m teaching double integrals, and there was this really great problem:

\underset{R}{\int\int}x\cos(xy)\cos^{2}(\pi x)dA

over the region R=[0,1/2]\times[0,\pi].

The problem is supposed to show that changing the order of integration can make the problem really easy or really hard.

\int_{0}^{1/2}\int_{0}^{\pi}x\cos(xy)\cos^{2}(\pi x)dydx is easy to solve.

\int_{0}^{\pi}\int_{0}^{1/2}x\cos(xy)\cos^{2}(\pi x)dxdy is hard to solve.

However, I don’t think the second integral should be impossible to solve. It’s been so long since I’ve really dug into integration, and we tried in class and couldn’t find a quick way to solve the second integral. Any ideas?

For those who know calculus but not Multivariable Calculus, the hard part of the second integral is simply being asked to solve: \int_{0}^{1/2}x\cos(cx)\cos^{2}(\pi x)dx, where c is simply a constant.

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3 comments

  1. @Michael: Ah, yes, at first I didn’t get what you meant, but then I realized you meant use \cos(a)\cos(b)=\frac{1}{2}(\cos(a+b)+\cos(a-b)) a couple times. I spent forever doing this, and it does seem to simplify things a bit. You also have to then use integration by parts to deal with \int x\cos(kx) dx. Then I stopped, but maybe I’ll finish it up later tonight to see if it does work out nicely.

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