# Multiple integral variable substitution using Jacobian matrix and matrix rotations

Question: By an appropriate choice of new variables evaluate the integral $\int\int_R(x^2+y^2)\,dx\,dy$ over the interior of the square bounded by $y=\pm x$ and $y=\pm (x-2)$.

I sketched the square region (symmetric about $x$-axis, with vertices $(0,0)$, $(1,-1)$, $(1,1)$, $(2,0)$).

I need to use the Jacobian matrix for this variable substitution (even if it is possible another way, this is the method I would like to use). I realise that an appropriate substitution would be to rotate the square by 45 degrees, either clockwise or anticlockwise. In order to do this I used the rotation matrix and get the following:

$$u=\frac{1}{\sqrt{2}}x+\frac{1}{\sqrt{2}}y;$$ $$v=\frac{1}{\sqrt{2}}y-\frac{1}{\sqrt{2}}x$$

Rearranging to get:

$$x=\frac{u-v}{\sqrt{2}}$$ $$y=\frac{u+v}{\sqrt{2}}$$

However, according to the answer sheet I have been given, at this stage I am already wrong and all $\sqrt{2}$ should actually just be $2$. Why is this?

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The answer sheet may use one particular transformation, but that doesn't mean all others are wrong. As long as you use the Jacobian correctly, you should be able to get the right result with this transformation. –  Muphrid Dec 29 '12 at 21:52