Find a solution to this integral analytically by calculating .
I am not sure about how to go about this question, how did they get ? Shouldn't ?
Thanks very much! (Just a bit confused on this question because I am working ahead of class )
I suspect you meant but that is also wrong. It is true that but that is NOT the same as . In fact, that last double integral makes no sense. You cannot integrate twice with respect to the same variable.
Remember that "x" is a dummy variable- the final answer will be a number so that it doesn't matter what you call the variable.
If then it is also true that or, for that matter, or .
We then can say that
That last part, saying that the product of the two integrals (each integral involving a different variable) is equal to the double integral is a version of "Fubini's theorem".
Of course, so we can write
The whole point of doing that is that now the integral is over the entire first quadrant and we can cast the integral into polar coordinates with going from 0 to and r from 0 to . Of course, in polar coordinates, and so we have
where I have used Fubini's theorem the other way to separate that double integral into a product of integrals.
Now that integral is easy and, because of the extra "r" so is the r integral.