Help with Changing of variables, Jacobian, Double Integrals?

[Suy]

Homework Statement

Show that T(u,v) = (u² - v², 2uv)
maps to the triangle = {(u,v): 0 ≤ v ≤ u ≤ 4} to the domain D
bounded by x=0, y=0, and y² = 1024 - 64x.

Use T to evaluate ∬_D sqrt(x²+y²) dxdy

Homework Equations

The Attempt at a Solution

x=u²-v²
y=2uv
Jacobian= 4u²+4v² dudv
I guess the equation in the changed variable integral should be ∫∫sqrt((u²-v²)²+(2uv)²) (4u²+4v²) dudv

But, I don't know how to get the bounds for the integrals in terms of u and v.
Can someone help me on this??

 

[sunjin09]

The first part of the problem already told you what the bounds of u and v are

 

[Suy]

I know that 0 ≤ u ≤ 4 and 0 ≤ v ≤4.
For y^2 = 1024 - 64x, x is restricted from 0 ≤ x ≤16 and 0 ≤ y ≤32??