Calculus: Coordinate Changes, Jacobian, Double Integrals?

[PinkPocky]

Calculus: Coordinate Changes, Jacobian, Double Integrals??

Homework Statement

Show that T(u,v) = (u² - v², 2uv)
maps to the triangle = { (u,v) | 0 ≤ v ≤ u ≤ 3 } to the domain D,
bounded by x=0, y=0, and y² = 324 - 36x.

Use T to calculate ∬sqrt(x²+y²) dxdy on the region D.

Homework Equations

The Attempt at a Solution

I know that dxdy = the Jacobian = (4u²+4v²)dudv.

I'm have a really hard time finding a way to figure what the bounds of the integral are, in terms of u and v.

 

[LCKurtz]

Homework Statement

Show that T(u,v) = (u² - v², 2uv)
maps to the triangle = { (u,v) | 0 ≤ v ≤ u ≤ 3 } to the domain D,
bounded by x=0, y=0, and y² = 324 - 36x.

Use T to calculate ∬sqrt(x²+y²) dxdy on the region D.

Homework Equations

The Attempt at a Solution

I know that dxdy = the Jacobian = (4u²+4v²)dudv.

I'm have a really hard time finding a way to figure what the bounds of the integral are, in terms of u and v.

Try drawing a picture of the uv region. You have 0 ≤ v ≤ u ≤ 3 given. Both u and v are between 0 and 3 so draw that square for a start. Now shade what part of that square also has v ≤ u. Then put your uv limits as that picture requires, like any other area problem.

 

[PinkPocky]

Great, thanks! I should have realized it before... but thank you, a really great explanation made it clear for me to help myself. :)

By the way, the final answer to the problem I solved for was ridiculous but correct. The solved integral is equal to 4536/5, or 907.2 :(... haha.