What is Elementary domains?

Example 20.12

On the triangle $D = \{0 \leq x \leq 1,\ 0 \leq y \leq x\}$:

Swapping the order ($x$ from $y$ to $1$): $\int_0^1 y\bigl(\int_y^1 x\,\dd x\bigr)\dd y = \int_0^1 y\,\frac{1 - y^2}{2}\,\dd y = \frac18$ — same value, different computation: choosing the order of integration well is half the craft.

Example 20.13 (When only one order works)

Compute $I = \displaystyle\int_0^1\!\!\int_x^1 \eu^{y^2}\,\dd y\,\dd x$. As written, the inner integral $\int\eu^{y^2}\dd y$ has no elementary antiderivative: the computation is stuck. But the domain is the triangle $0 \leq x \leq y \leq 1$, which is elementary in both directions; swapping the order,

The inner variable $x$ appeared nowhere in the integrand, so integrating it first produced exactly the factor $y$ that makes the outer integral immediate. Moral: Fubini is not just a license to iterate — it is a license to choose, and the right order can turn an impossible integral into a one-liner. Always sketch the domain and read off both descriptions before starting.

Example 20.18 (Affine changes of variables)

For an affine map $\Phi(u, v) = M(u, v)^{\mathsf T} + C$ with $M$ invertible, the Jacobian is the constant matrix $M$: areas are multiplied by the constant factor $\abs{\det M}$ — the promise made in Chapter 17 is now a theorem. Two immediate uses. The ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1$ is the image of the unit disk under $(u, v) \mapsto (au, bv)$, so its area is $ab \cdot \pi$ — no computation. And for the integral of $f(x + y)$ over the square $K = \intcc01^2$, the shear $\Phi(u, v) = (u - v, v)$ (determinant $1$) turns it into an integral of $f(u)$ over a parallelogram, which Fubini slices at constant $u$: with $f = \exp$,

as the product structure confirms. Choosing coordinates adapted to the integrand — not to the domain — is the other half of the craft.