Fubini's theorem

Fixing \(x\) and computing \(A(x)=\int_c^d f(x,y)\,dy\) measures the area of the slice of the solid above the vertical line at that \(x\). Integrating \(A(x)\) from \(a\) to \(b\) adds up all the slices and gives the volume. This is Cavalieri's principle, and it is what the theorem says: the volume does not depend on whether you slice parallel to one axis or the other.

In practice you integrate from the inside out: in the inner integral, the other variable is treated as a constant. Both orders give the same number, but one can be much simpler; for \(x e^{xy}\), for instance, it pays to integrate in \(y\) first, since an antiderivative in \(x\) would require integration by parts.

When \(f(x,y)=g(x)h(y)\) on a rectangle, the double integral factors: \(\iint_R f\,dA=\int_a^b g\,dx\cdot\int_c^d h\,dy\). Guiding example: \(\iint_R xy\,dA\) with \(R=[0,2]\times[0,3]\) equals \(2\cdot\tfrac92=9\). This factoring is valid only on rectangles and for functions that are a product of a function of \(x\) and a function of \(y\).

Typical mistakes: factoring when \(f\) is not a product (for instance \(x+y\)), forgetting that each integral's limits belong to its own variable, and applying the theorem without continuity. Regions that are not rectangles are handled in the double integral over a general region.