1
$\begingroup$

I was trying to evaluate double integral: $\int_{0}^{\pi}dx\int_{x}^{\pi} \frac{\sin y}{y} dy$

I don't know what to do, from double integral calculator the answer is $2$. I checked indefinite integral from it and it is $\operatorname*{Si}(x)+C$. I tried to do it with polar form, but I get nothing interesting.

How to evaluate such integral?

$\endgroup$
  • 2
    $\begingroup$ Try to change the order of integration (allowed, because the integrand is positive in that area). $\endgroup$ – Professor Vector Jun 11 '17 at 15:38
4
$\begingroup$

Switching the order of integration

$$ \int_{0}^{\pi}\mathbb{d}x\int_{x}^{\pi}\mathbb{d}y=\int_{0}^{\pi}\mathbb{d}y\int_{0}^{y}\mathbb{d}x $$

your integral is then

$$ \int_{0}^{\pi}\mathbb{d}y\int_{0}^{y}\frac{\sin(y)}{y}\mathbb{d}x=\int_{0}^{\pi}\sin(y)\mathbb{d}y=2 $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.