double integrals

**bille** · January 16th 2011, 07:41 AM

Could someone go through this step step by step?
I get stuck at part ii

Thanks

**Opalg** · January 16th 2011, 09:45 AM

Originally Posted by bille

Could someone go through this step step by step?
I get stuck at part ii

Thanks

Assuming that you have done part (i), the essential next step is to draw a diagram of the domain D. That should enable you to describe D in terms of u and v, namely that D is enclosed by the lines $u=1$ , $u=e$ , $v=0$ and $v=\pi$ . The change of variables formula then tells you that the integral is equal to

. . . . . $\displaystyle\int_1^e\int_0^\pi\frac{\sin v}{x^2}\,\frac{x^2}u\,dvdu$ .

The " $x^2$ "s conveniently cancel, and from then on the integral should be easy.

**bille** · January 16th 2011, 09:48 AM

Sorry, what is the change of variable formula?
Thanks for our reply.

**Opalg** · January 16th 2011, 10:10 AM

Originally Posted by bille

Sorry, what is the change of variable formula?

The formula says that if you make a change of variables from (x,y) to (u,v), so that f(x,y) becomes g(u,v), then the integral $\displaystyle\iint_Df(x,y)\,dxdy$ becomes $\displaystyle\iint_Dg(u,v)\left|\frac{\partial(x,y )}{\partial(u,v)}\right|dudv$ .

See here for more on this topic. (But if you are being asked to do problems like this, then presumably it is expected that you already know this stuff.)

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