Calculate $\iint\limits_D (x+3y)*(-2x+y)-x-3y dxdy$ where D is the rectangle with corners at (4, 1), (5, 3), (-5, 4) and (-4,6) I am trying to calculate the double integral $\iint\limits_D (x+3y)*(-2x+y)-x-3y dxdy$ where D is the rectangle with corners at (4, 1), (5, 3), (-5, 4) and (-4,6).
I started by sketching the area and realized that I probably have to divide the integration into two areas so that D=D_1 + D_2. I also calculated the linear equations connecting the points because I'm thinking that these lines are going to be the upper and lower bounds for the integrals, is that right? However I am really new to double integrals so I'm thinking this might be wrong or a very complicated way of solving this. I also don't get a volume when setting these linear equations as my bounds, I just integrate and switch x for y and vice versa, which does not seem right. Does anyone have any input? If so, I'd be very grateful. You can see my calculations and reasoning below:
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Thanks in advance!
/Nick
 A: If points of the rectangle are $A(4,1), B(5,3), C(-4,6), D(-5,4)$, please note that once you find the slope of $AB$, $CD$ will have the same slope. Similarly once you find for $BC$, you know it for $DA$. Also, when writing equations of lines note that two adjacent sides of rectangle go through a common point. These are small things but just speeds things up for you a bit.
$AB: (y-1) = 2(x-4), DA: (y-1) = - \frac{1}{3}(x-4)$
$CD: (y-6) = 2(x+4), BC: (y-6) = -\frac{1}{3}(x+4)$
Simplifying, we have a rectangle bound by the below $4$ lines,
$-7 \leq - 2x + y \leq 14, 7 \leq x + 3y \leq 14$
Now if you want to do this integral without change of variable, it is $3$ integrals. But if we look at the integrand and the bounds, it is pretty clear that we do change of variable as $u = - 2x + y, v = x + 3y$.
The Jacobian of transformation, $|J| = \frac{1}{7}$ and here are our new bounds -
$- 7 \leq u \leq 14, \ 7 \leq v \leq 14$.
Integrand $(x+3y) (-2x+y)-x-3y$ becomes $(uv - v)$
The integral becomes
$I = \displaystyle \frac{1}{7} \int_{7}^{14} \int_{-7}^{14} (uv - v) \ du \ dv$
