use the transformation T(u,v) = < u, u+v> to evaluate the iterated integral
the integral from 1 to 2 the integral from x+3 to x+6 of 1/(sqrt(xy-x^2)) dy dx.
please help if you know how to solve this question. thank you.
I will attach the graphs later if i have time. i will show you how to set up the integral.
Recall how we perform change of variables with Jacobians:
Assuming all the conditions are fulfilled (look up these conditions in your text), we calculate the integral under the transformation T as follows:
where
we are given:
and
So, we have:
Now, graph the region over which the original integral is being done, it will be a parallelogram. label the lines as i describe so you can follow, we have to transform each line under the directions of T. Label the line y = x + 3 as , the line x = 2 as , the line y = x + 6 as , and the line x = 1 as . note the intervals on which these lines are defined, as we have to change them under the transformation T as well. ( , , , and )
Now, let's get this show on the road.
For
For
For
For
Now we are ready. draw all those graphs between the given intervals on a new pair of axis. Let the vertical axis be and the horizontal axis be . you will notice the new region is a rectangle, which is a much simpler region to integrate over. not only that, but the integral gets a lot easier as well:
We know:
Now continue
Check my computations, i only got 2 hours of sleep last night, so i am really tired and prone to silly mistakes right now
EDIT: I attached the diagrams of the regions