Transformations in double integrals. (Jacobian)

[VinnyCee]

Evaluate

$$\int\int_{R} \left(2x^2 - xy - y^2\right) dx\;dy$$

by applying the transformation

$$u = x - y , v = 2x + y$$

for the region R in the first quadrant bounded by the lines

$$y = -2x + 4, y = -2x + 7, y = x - 2, y = x + 1$$

I don't even know where to start! Please help.

 

[VinnyCee]

I have this so far

$$y = x - u$$

$$v = 2x + x - u$$

$$v = 3x - u$$

$$3x = v + u$$

$$x = \frac{1}{3} \left(u + v\right)$$

$$y = \frac{1}{3} \left(-2u + v\right)$$

then, using partial derivatives to get the Jacobian to be $$\frac{1}{3}$$, right?

Then I use those equations and the line equations to solve for the limits of u and v:

$$u = -1\;to\;2$$

$$v = 4\;to\;7$$

The resulting transformed integral (with Jacobian in front)would then be:

$$\frac{1}{3}\int_{-1}^{2}\int_{4}^{7} v\;u\;dv\;du = \frac{33}{4}$$

Is this correct?

 

[saltydog]

$$\frac{1}{3}\int_{-1}^{2}\int_{4}^{7} v\;u\;dv\;du = \frac{33}{4}$$

Is this correct?

I think you need to transform the integrand in terms of u and v, that is:

$$\iint_R F(x,y)dxdy=\pm\iint_S F(f(u,v),g(u,v))\frac{\partial (x,y)}{\partial (u,v)}dudv$$

Note the plus/minus sign.

 

[VinnyCee]

Is this part correct?

$$\frac{\partial (x,y)}{\partial (u,v)} = \frac{1}{3}$$

But what is this equal to, how do I find it?

$$F(f(u,v),g(u,v)) =\;?$$

 

[saltydog]

The Jacobian is correct. It's just a 2x2 determinant (1/6+1/6) right?

To transform the integrand, just substitute the expressions for x and y in terms of u and v. You know, x=1/3(u+v) and y=1/3(v-2u). So $x^2$ is . . . xy is . . . and so on. Then integrate over the transformed area in the u-v plane which is a nice square.

 

[VinnyCee]

After your advice, i get the following

$$\frac{1}{3}\;\int_{-1}^{2}\int_{4}^{7} \left(-\frac{1}{9}\;u^2 - \frac{1}{9}\;v^2 + \frac{7}{9}\;u\;v\right)\;dv\;du$$

Is this right?

 

[saltydog]

Well, I didn't simplify it but for:

$$2x^2-xy-y^2$$

with:

$$x=\frac{1}{3}(u+v)$$

$$y=\frac{1}{3}(v-2u)$$

Isn't that just:

$$2[\frac{1}{3}(u+v)]^2-\frac{1}{9}(u+v)(v-2u)-\frac{1}{9}(v-2u)^2$$

 

[VinnyCee]

Simplified and hopefully I am understanding the concept now!

This

$$2[\frac{1}{3}(u+v)]^2-\frac{1}{9}(u+v)(v-2u)-\frac{1}{9}(v-2u)^2$$

simplifies to

$$-\frac{4}{9}\;u^2 + \frac{2}{9}\;v^2 + \frac{7}{9}\;u\;v$$

(I think )

Now that the Jacobian has been checked and we just simplified the re-substitution (correct term? ), are the limits of integration correct?

$$\frac{1}{3}\;\int_{-1}^{2}\int_{4}^{7} \left(-\frac{4}{9}\;u^2 + \frac{2}{9}\;v^2 + \frac{7}{9}\;u\;v\right)\;dv\;du$$

Does everything seem to be in order?

 

[saltydog]

This

$$2[\frac{1}{3}(u+v)]^2-\frac{1}{9}(u+v)(v-2u)-\frac{1}{9}(v-2u)^2$$

simplifies to

$$-\frac{4}{9}\;u^2 + \frac{2}{9}\;v^2 + \frac{7}{9}\;u\;v$$

(I think )

Now that the Jacobian has been checked and we just simplified the re-substitution (correct term? ), are the limits of integration correct?

$$\frac{1}{3}\;\int_{-1}^{2}\int_{4}^{7} \left(-\frac{4}{9}\;u^2 + \frac{2}{9}\;v^2 + \frac{7}{9}\;u\;v\right)\;dv\;du$$

Does everything seem to be in order?

Vinny, sorry I got to you late. Probably you've figured it out by now but I get just uv for the integrand:

$$\frac{1}{3}\;\int_{-1}^{2}\int_{4}^{7}(uv)dvdu=\frac{99}{12}$$

 

[mkkrnfoo85]

just wondering

Just wondering... I'm learning the change of variables Jacobian things too right now and also having some problems as well.

Would someone tell me how to find the limits of integration for 'u' and 'v'?
Like you got:

u = -1 to 2
v = 4 to 7

How do you find those?

 

[Oggy]

The area is bounded by y=-2x+4 and y=-2x+7, so that's y+2x=4 and y+2x=7, or v=4 to v=7, so v goes from 4 to 7, and the same for u. With this change of variables, you're actually fitting the area of integration to a rectangle.

 

[mkkrnfoo85]

yea. that makes so much sense now, putting it into a rectangle like that. thanks.