25 points scalc9m 159025mi evaluate the given integral by...

[(-)/(2.5) Points] SCALC9M 15.9.025.MI. Evaluate the given integral by making an appropriate change of variables. ∬_(R)7(x-4y)/(8x-y)dA, where R is the parallelogram enclosed by the lines x-4y=0,x-4y=7,8x-y=6, and 8x-y=9 Need Help?

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00:02 We're going to evaluate the given integral by making appropriate change of variables.

00:06 So notice that the lines that define the parallelogram over which we're integrating are x minus 4y as a constant and 3x minus y is a constant.

00:16 Those are also the lines involved in the integrands.

00:18 So those are good options for me to make my new variables.

00:22 I'm going to let w be x minus 4y and i'm going to let z be 3x minus y.

00:31 Then i'm integrating from w is 0 to w is 8, from z is 3 to z is 9, and the integrand is just w over c.

00:46 Now to find the integrating factor here, i need the jacobian, so i need to be able to calculate the derivative of x with respect to w and with respect to c, and then also the derivative of y with respect to w.

01:02 And with respect to z.

01:05 So with that, i need to write x and y in terms of c and w.

01:09 Okay.

01:10 So x is going to be w plus 4y.

01:14 Putting that in for x over here, i get z is equal to three times w plus 4y minus y.

01:23 So z is equal to 3w plus 12y minus y.

01:29 So z minus 3w is 11y.

01:33 So z over 11 minus 3 over 11 w is y.

01:40 And putting that for x is going to be w plus 4 times z over 11 plus no minus 3 over 11 w.

01:54 So that is w plus 4 over 11 z minus 12 over 11 w.

02:01 So x is w negative 1 over 11 w plus 4 over 11c...