Hello everyone, I am having difficulty with this double integral.

I was given the following integral:
\( I =\int ^1_0 dx \int ^{2x}_x 2xy dy \)

I have to reverse the order of integration and so this was my approach:

I first plotted the graph of the above and is attached as 'integrate along y'.

So then I integrated along x so the lower limit is y and y/2. The second image to show what i was thinking is shown as 'figure 2'.

So the integral that I get is:
\( I =\int ^1_0 dy \int ^{y}_{y/2} 2xy dx \)

However, my solution comes to 3/16 whereas the answer should be 3/4.

Thanks fore reading and hopefully one could explain what I am doing wrong!

 

The region you are integrating over is bounded by three lines, not two.

There is y=x, y=2x and x=1. They form a triangle shape. Recheck your limits and you will see that your new integral does not span over the same triangular region.

 

Thanks steveb. One other issue that I am having is the following. For the first integral that I wrote, the limits for the first part of the integral are 0 and 1. I used the second integral limits to arrive at the two equations (y=x and y=2x) and from previous examples, i would combine the two together to arrive at the limits for the first integral. This does not work in this instance.

The reason why I am asking this is because I am confused what to put for the first integral for when I reverse the order of integration.

 

Actually, that was a silly question, because the lines do not cross!! Please ignore my above post