Calculating Int. ∫D (6x+12y) dA using Change of Variables

[shards5]

Homework Statement

Suppose D is the parallelogram in the xy-plane with vertices P(-1,5), Q(1,-5), R(5,-1), S(3,9)
$$\int\int ^{}_{D} (6x+12y) dA$$
HINT: Use transformation x = $$\frac{1}{6}$$(u+v) and y = $$\frac{1}{6}$$ (-5u+v).

Homework Equations

The Attempt at a Solution

Calculating the Jacobian I get
dx/du = 1/6 dx/dv = 1/6
dy/du = -5/6 dy/dv = 1/6
which gives me
1/36 - (-5/36) = 1/6
From there I need the new intervals of integration.
P-Q
$$\frac{5-(-5)}{-1-(1)}$$ = 10/-2 = -5
y = 5x + 5 -> plugging in . . .
-5/6u+1/6v = -5/6u - 5/6v + 5 -> u cancels out and then we get 6/6v = 5 which means v = 5
S-R
$$\frac{9-(-1)}{3-5}$$ = -5
y = -5x + 9
-5/6u+1/6v = -5/6u+5/6v + 9 -> v = 9
R-Q
$$\frac{(-1)-(-5)}{5-1}$$ = 1
y = x-1 -> -5/6u + 1/6v = 1/6u +1/6v -1 -> u = 1
S-P
$$\frac{(9-(5)}{3-(-1)}$$ = 1
y = x-9 -> -5/6u +1/6v = 1/6u + 1/6v - 9 -> u = 9

Plugging in the x and y values and the Jacobian into the integral we get the following new integral.
$$\int^{9}_{5}\int^{9}_{1} (u+v + (-10u+2v)) * 1/6 dudv$$
which simplifies to . . .
$$\int^{9}_{5}\int^{9}_{1} (-4/3u+1/2v) dudv$$
After the first integration I get . . .
$$\int^{9}_{5} (-2/3u^2+1/2v*u) dudv$$
Plugging in 9 and 1 I get
-54-9/2v+50/3-5/2v = -37.33333 - 7v
Integrating what I got I get
-37.33333v - 7/2v²
And after plugging in 9 and 5 I get -345.333
which is wrong. So what am I doing wrong?

 

[mighty2000]

There are a few mistakes in your solution. First, when calculating the Jacobian, you need to use the absolute value of the determinant, so it should be 1/6 + 5/6 = 2/3, not 1/6 - (-5/6) = 1/6 + 5/6 = 2/3.

Secondly, when finding the new intervals of integration, you need to consider both the x and y values, not just one. For example, for P-Q, you correctly found that the slope is 5, but you also need to use the point (1,-5) to find the equation of the line, which is y = 5x - 10. Similarly, for S-R, the slope is -5, but the point (5,-1) gives the equation y = -5x + 24.

Finally, when plugging in the Jacobian and the new intervals of integration into the integral, you should have (-4/3u + 1/2v + 6u + 2v) instead of (-4/3u + 1/2v + (-10u + 2v)).

Correcting these mistakes should give you the correct answer.