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I'm having trouble with this problem: Evaluate the following double integral

$$ \iint_D (x^2+y^2) \,\text d x\,\text dy $$

where $D$ is the region comprised of a parallelogram with the corners $(0,0)$, $(-1,1)$, $(4,2)$ and $(3,3)$.

If I look at the function, it seems easier to solve with polar coordinates, but if I look at the region, it seems easier to do in the cartesian plane. The cartesian plane makes the calculation very complex with many steps and I seem to always make a mistake when trying it.

I've figured that the region is comprised of the lines $A: x=2y$, $B: x= 2y-3$, $C: x=-y$ and $D: x=-y+6$.

I would transform the function to polar form as follows:

$$ \iint_Dr^2 \, \text dr \, \text d\theta $$

but how do I switch the integral bounds to polar form? I tried to simply substitute $x = r\cos\theta$ and $y=r\sin\theta$ for the lines $A$, $B$, $C$ and $D$; but I don't know if I should try solving for $\theta$ or for $r$, and I don't feel very confident with the trigonometry when I test this. Maybe I can use the Jacobian matrix to make it easier for me, but I haven't used the Jacobian matrix in practice before, so I feel a bit lost here.

  • is this integral easier with polar coordinates?
  • how do you change to polar coordinates for these bounds?
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    $\begingroup$ ...change the parallelogram into a rectangle by a linear transformation, no polar coordinates. $\endgroup$ – dan_fulea Aug 8 '18 at 12:42
  • $\begingroup$ You are computing the moment of inertia of a parallelogram with respect to a corner. You may compute the moment of inertia with respect to the centroid in a simple way (it is enough to recall the moment of inertia of a rectangle with respect to its centroid), then invoke the parallel axis theorem. $\endgroup$ – Jack D'Aurizio Aug 8 '18 at 17:40
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HINT

We can set

  • $u=x+y \implies 0\le u\le 6$

  • $v=2y-x\implies 0\le v\le 3$

then

$$\iint_D (x^2+y^2) \,dx\, dy=\iint_R \frac13 \left[\left(\frac{2u-v}3\right)^2+\left(\frac{u+v}3\right)^2\right] \,du\, dv$$

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