Changing the order of integration

jeepers101

I have no idea how to write the question so I've scanned and attached my attempt.

The original integral, as I understand it is integrating from y=sinx to y=3sinx from o to π. (Vertical arrow)

Now I want to integrate in the direction of the horizontal arrow.

My problem is that in my graph, above y=1, I cannot integrate from x=arcsin(y/3) to x=arcsiny. I am just integrating from x=arcsin(y/3) to itself.

I suspect I need to break the graph into more than one section and present the solution as the sum of these sections but I don't know how to do this.

Any help is much appreciated.

jeepers101

I think I have solved it.

I have broken the graph into four sections. Two of the sections are to the right of x = π/2 and the other two are to the left. Because of the symmetry I can just concentrate on the left hand side and double the answer.

So now the first section I will integrate from x=arcsin(y/3) to x=arcsiny between 0 and 1.

For the second section I will integrate from x=arcsin(y/3) to x=π/2 between 1 and 3.

Now as I said already, double the sum of these integrals.

* π is pi.