Calculus: Coordinate Changes, Jacobian, Double Integrals?

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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.
 

Answers and Replies

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LCKurtz
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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.
 
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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.
 

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