Fubini's Theorem

If a function is Lebesgue integrable on a rectangle $X\times Y$, then the double integral can be evaluated by an interated integral ${\iint }_{X\times Y}f(x,y)d(x,y)={\int }_X\left({\int }_{Y}f(x,y)dy\right)dx={\int }_Y\left({\int }_{X}f(x,y)dx\right)dy$

If a function is continuous on the rectangle, the theorem applies to the Riemann Integral.