Using Jacobian to determine area

[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]

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?

 

[HallsofIvy]

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

 

[snoggerT]

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

- 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.

 

[HallsofIvy]

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.

 

[snoggerT]

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.

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

 

[HallsofIvy]

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.