Using Jacobian to determine area

1. Jul 10, 2008

snoggerT

Let [phi](u,v)=(3u+v,u-2v). Use the Jacobian to determine the area of [phi]R for:

R=[2,5]X[1,7]

3. The attempt at a solution

- I'm really not sure why I keep getting the wrong answer on this problem. the problem gives you two R's to solve for and I got the right answer for the first one (R=[0,3]X[0,5]), but I'm not getting the right answer for the 2nd R. I would think you would solve the problems the exact same way, but I'm not sure since I can't get the right answer. Would R being changed affect how you work the problem?

2. Jul 10, 2008

HallsofIvy

Staff Emeritus
Well, what did you do? What is the Jacobian? And what answer did you get?

3. Jul 10, 2008

snoggerT

- my jacobian was |-7|. I set the limits on my outer integral (for dv) from 1 > 7 and my inner integral from 2 > 5. I integrated with respect to u first and then integrated with respect to v. I kept getting a very large negative number, but the answer is positive and not that big.

4. Jul 10, 2008

HallsofIvy

Staff Emeritus
How could you possibly have gotten negative answer? The Jacobian is, as you say, 7, and the area of the rectangle is (7-1)(5-2)= 6(3)= 18 so the area of the transformed region would be 18(7)= 126.

5. Jul 10, 2008

snoggerT

- So you don't use integration on this problem? You can just multiply the area of the rectangle times the jacobian?

6. Jul 10, 2008

HallsofIvy

Staff Emeritus
that's the whole point of the problem! No you don't need to use integration to find the area of a rectangle and the point of the Jacobian is that it changes the area of one region to the area of the other.