U substitution and integration

[3soteric]

Homework Statement

use the substitution u= x+y and v=y-2x to evaluate double integral from
∫1-0∫(1−x) -(0) of (√x+y) (y−2x)^2 dydx

Homework Equations

integration tables I am assuming

The Attempt at a Solution

i tried to integrate directly but none of my integration tables match up to the format

 

[clamtrox]

You should do as the assignment tells you to do.

So perform the change of variables first.

 

[Millennial]

I will write it in latex for those who want to solve it:
$$\int^{1}_{0} \int^{1-x}_{0} \sqrt{x+y} (y-2x)^2\,dy\,dx$$
Also, the Jacobian of the transformation you are trying to perform is
$$\begin{vmatrix} 1 & 1 \\ -2 & 1 \end{vmatrix}$$
What does that equal?

 

[3soteric]

the jacobian equals 3 but how is that related to the entire problem ? :s

 

[SammyS]

Homework Statement

use the substitution u= x+y and v=y-2x to evaluate double integral from
∫1-0∫(1−x) -(0) of (√(x+y)) (y−2x)^2 dydx

Homework Equations

integration tables I am assuming

The Attempt at a Solution

i tried to integrate directly but none of my integration tables match up to the format

the Jacobian equals 3 but how is that related to the entire problem ? :s

You need the Jacobian to change dy dx to du dv or dv du .

You will also need to change the limits of integration.

Solving the system of equations,

u= x+y

v=y-2x​

for x & y, will help you to do that.

Sketch the region of integration for the given integral, $\displaystyle \int^{1}_{0} \int^{1-x}_{0} \left(\sqrt{x+y\ }\, (y-2x)^2\right)\,dy\,dx\,,\$ in the xy-plane. Then convert that to the corresponding region in the uv-plane.