Could someone go through this step step by step?

I get stuck at part ii

http://img814.imageshack.us/img814/2227/mathse.jpg

Thanks

Printable View

- Jan 16th 2011, 07:41 AMbilledouble integrals
Could someone go through this step step by step?

I get stuck at part ii

http://img814.imageshack.us/img814/2227/mathse.jpg

Thanks - Jan 16th 2011, 09:45 AMOpalg
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 $\displaystyle u=1$, $\displaystyle u=e$, $\displaystyle v=0$ and $\displaystyle v=\pi$. The change of variables formula then tells you that the integral is equal to

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

The "$\displaystyle x^2$"s conveniently cancel, and from then on the integral should be easy. - Jan 16th 2011, 09:48 AMbille
Sorry, what is the change of variable formula?

Thanks for our reply. - Jan 16th 2011, 10:10 AMOpalg
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 $\displaystyle \displaystyle\iint_Df(x,y)\,dxdy$ becomes $\displaystyle \displaystyle\iint_Dg(u,v)\left|\frac{\partial(x,y )}{\partial(u,v)}\right|dudv$.

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.)