Question

Double Integrals: Want to make sure I've evaulated this integral correctly.

y/(1+x^2)dydx

Answer 1

\[\int\limits_{}^{}\int\limits_{}^{}y/(1+x^2)dydx\]

I was also given limits and asked to find one but I just want to focus on evaulating the integral

Answer 2

Actually, I'll give you guys the limit:
\[\int\limits_{y^2}^{4}\int\limits_{0}^{2}(y/(1+x^2)dydx\]

Answer 3

the y^2 was originally \[y=\sqrt{x}\]

Answer 4

\[ ∫∫y/(1+x^2)dydx = ∫y dy∫dx/(1+x^2)\\
= \frac {y^2}2 \arctan(x) + C
\]

Answer 5

\[\int\limits_{y^2}^{4}1/2(y^2 \arctan(x)_{0}^{2}dx\]
\[\int\limits_{y^2}^{4} 2\arctan(x)dx\]

Answer 6

yeah I got that as well

Answer 7

^ then after plugging in the limits and evaluating with respect to dy I get what I have above

Answer 8

From there I I'm not sure what the integral of arctanx would be, I think its:
\[1/2\log(x^2+1)\]

Answer 9

Is there any reason to have a variable in the limits of the outer integral? From what I remember, that shouldn't be: a defined integral should result in a number. How about switching the order of integration?
Well, I am sorry if I didn't understand correctly what you wrote. I am quite sleepy now :-)

Answer 10

Also, integral of arctan(x) = xarctan(x) - (1/2)log(x^2 + 1) (Integral by parts.)

Answer 11

^ I'm honestly not sure.

I figured out that mistake not too long ago referring to your answer