Question

How to evaluate this double integral

Answer 1

\[\int\limits_{0}^{1}\int\limits_{0}^{y}\sqrt{4y^2+5}~dx~dy\]

Answer 2

Is there anything you can do to it besides trig sub?

Answer 3

I'll be right back

Answer 4

I think that your integral can be rewritten as below:
\[\huge \int_0^1 {dy} \sqrt {4{y^2} + 5} \int_0^y {dx} \]

Answer 5

your integrand doesn't depend on x...
you are first basically integrating a constant w.r.t. x which is just that constant*x

\[\int\limits_0^y c dx=cx|_0^y=c(y-0)=cy \\ \text{ where the constant in question is } \sqrt{4y^2+5}\]

Answer 6

Yes, I'll go along with that

Answer 7

then you move to outside integral which just requires a simple substitution

Answer 8

No trig sub required because we end up with the integral

\[\int\limits_{0}^{1}2y \sqrt{(y^{2}+5/4} ) dy\]

Answer 9

Oh i see. Thank you!

Answer 10

I didn't notice it was supposed to be a constant haha