Double Integral

Thread Starter


Joined Mar 22, 2011
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!



Joined Jul 3, 2008
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.

Thread Starter


Joined Mar 22, 2011
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.