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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 )
There is a typo in your final line- you surely don't want . Where did that final "dy" come from?
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.