# Understanding limits of integration after transformation $(x,y) \mapsto (x-y,x+y)$

Consider the following double-integral:

$$\iint_{[0,a]\times[0,a]} (x-y)^2 dxdy \stackrel{*}{=} \int_0^a \int_0^a (x-y)^2 dxdy = \frac{a^4}{6}$$

where $$*$$ follows by iterated integration. Suppose instead we consider the substitution $$w=x-y, z=x+y$$. The Jacobian of this transformation is $$1/2$$, and so

$$\iint_{[0,a]\times[0,a]} (x-y)^2 dx dy = \frac{1}{2} \iint_{R_2} w^2 dw dz =\frac{1}{2} \int_{Y_1}^{Y_2} \left (\int_{X_1}^{X_2} dz\right )w^2 dw.$$ My question is: how do I go about solving this via iterated integration (with respect to $$z$$ first, then with respect to $$w$$), or more specifically, what should the limits of integration $$Y_1,Y_2,X_1,X_2$$ be?

If we consider the transformation $$(x,y)\mapsto (w,z)$$ as a linear map

$$A = \begin{bmatrix}1 & -1 \\ 1 & 1\end{bmatrix}$$

then this transforms the square $$[0,a]\times [0,a]$$ into the parallelogram with vertices: $$(0,0), (a,a),(0,2a), (-a,a)$$, which is the region bounded by the lines: $$y=x, \quad y=x+2a, \quad y=-x+2a, \quad y=-x,$$

or equivalently by the lines $$w=0, \quad w=-2a, \quad z=0, \quad z=2a,$$ and so $$R_2 = \{(w,z) : -2a \le w \le 0, ~ 0\le z \le 2a \}.$$ This is not a rectangular region in $$(w,z)$$-space, but clearly it is not correct to take $$Y_1 = -2a, Y_2=0, X_1 = 0, X_2=2a$$, since it does not lead to the answer found by direct integration mentioned in the first display. My confusion is about how to relate $$z$$ and $$w$$ and choose the inner integral limits properly.

• Your first double integral needs $dxdy$. Commented Jan 17 at 22:07

So, $$w$$ can take any value from $$-a$$ to $$a$$ and
• if $$w\in[-a,0]$$, $$z$$ can take any value from $$-w$$ to $$w+2a$$;
• if $$w\in[0,a]$$, $$z$$ can take any value from $$w$$ to $$2a-w$$
So, your integral is$$\frac12\left(\int_{-a}^0\int_{-w}^{w+2a}w^2\,\mathrm dz\,\mathrm dw+\int_0^a\int_w^{2a-w}w^2\,\mathrm dz\,\mathrm dw\right),$$which turns out to be equal to $$\dfrac{a^4}6$$ (as it should be).
• would it possible to elaborate on how you decided to split the possible values of $w$ and how you found the bounds for $z$? I'm still not quite seeing the picture here Commented Jan 17 at 21:46