Help with Changing of variables, Jacobian, Double Integrals?

In summary, the problem asks to evaluate the integral of the function sqrt(x^2+y^2) over the domain D, which is bounded by the lines x=0, y=0, and y^2=1024-64x. This domain is mapped to the triangle {(u,v): 0 ≤ v ≤ u ≤ 4} by the transformation T(u,v) = (u^2-v^2, 2uv). The Jacobian is 4u^2+4v^2, and the bounds for the integral in terms of u and v are 0 ≤ u ≤ 4 and 0 ≤ v ≤ 4. The bounds for x and y are 0 ≤ x ≤
  • #1
Suy
101
0

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??
 
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  • #2
The first part of the problem already told you what the bounds of u and v are
 
  • #3
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??
 

FAQ: Help with Changing of variables, Jacobian, Double Integrals?

1. What is a change of variables in double integrals?

A change of variables in double integrals is a technique used to simplify the calculation of integrals by transforming the original coordinates to new coordinates. This can be done to make the integral easier to evaluate or to change the shape of the integration region.

2. What is the Jacobian in a double integral?

The Jacobian in a double integral is a mathematical concept that represents the ratio of the area in the original coordinates to the area in the new coordinates. It is used to account for the change in scale and orientation when transforming the integration region.

3. Why is the Jacobian important in double integrals?

The Jacobian is important in double integrals because it allows us to account for the change in scale and orientation when transforming the integration region. This ensures that the integral is calculated correctly in the new coordinates.

4. How do you find the Jacobian in a double integral?

The Jacobian in a double integral can be found by taking the derivative of the transformation functions with respect to the new coordinates and then taking the determinant of the resulting matrix. This gives us the Jacobian in terms of the new variables.

5. What are some common applications of changing variables and the Jacobian in double integrals?

Changing variables and the Jacobian are commonly used in various fields such as physics, engineering, and economics to solve integrals in different coordinate systems. They are also used in probability and statistics to transform probability distributions and in image processing to change the orientation and scale of images.

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