Reversing Order of Integration for Double Integral Problem

[emelie_earl]

My idea was that the limits are

and that the anti-derivative of dy was

xlog(1+y^2)

but that seems wrong...

maybe use these limits instead

and start with dx?

gives us

then we take dy

guess, i figured it out eventually with the help of wolfram with the last integration

 

[SammyS]

The reason your 1st attempt didn't work is that $$\int\frac{1}{1+y^2}\,dy=\tan^{-1}(y)+C\,.$$

 

[HallsofIvy]

My idea was that the limits are

and that the anti-derivative of dy was

xlog(1+y^2)

but that seems wrong...

Yes, it is wrong. It is standard "Calculus I" mistake to treat a function of the variable as if it were just the variable but you should be past that by the time you are doing multiple integrals. "$1/(1+ y^2)$" is NOT the same as $1/y$ and its anti-derivative is not a logarithm. The anti-derivative of $1/(1+ y^2)$ is $arctan(y)+ C$. That's a standard anti-derivative that you should have memorized.

maybe use these limits instead

and start with dx?

gives us

then we take dy

guess, i figured it out eventually with the help of wolfram with the last integration

Yes, reversing the order of integration is the best way to handle this one. Integerating $arctan(1)- arctan(x^2)= \pi/4- arctan(x^2)$ is likely to be very difficult!

 

[SammyS]

...
Yes, reversing the order of integration is the best way to handle this one. Integerating $arctan(1)- arctan(x^2)= \pi/4- arctan(x^2)$ is likely to be very difficult!

Yes. I agree !