so Ive just been introduced to the idea of iterated integrals and im finding it hard to work out how to complete questions on this subject and was wondering if anyone could help.
So if I have an intial question, say:
$\int_{0}^{1}\int_{0}^{y^5}(\pi sin(5\pi x)/(1-x^{1/5}) dxdy$
I know that I have to intially look at the inequalities of the limits which I think is:
$0\leq y\leq1$ and $0\leq x\leq y^5$
And I believe that I have to draw the region defined by the inequalities but I'm not entirely sure how to do that.
From there I think I have to described the drawn region by the new inequalities but with the variables in the opposite direction. And once I have defined them, put the new inequalities back onto the integral and do the process of double integration.
So hence I think I know what I am doing with the beginning and the latter but I am stuck on the process in the middle.
Any help would be greately appreciated.
If you change the order of integrations you get
$$\int_{0}^{1}\int_{0}^{y^5}(\pi \sin(5\pi x)/(1-x^{1/5}) dxdy=\int_{0}^{1}\int_{x^{1/5}}^{1}(\pi \sin(5\pi x)/(1-x^{1/5}) dydx.$$
This gives the integral
$$\int_{0}^{1}(\pi \sin(5\pi x) dx.$$