Fubini's Theorem

If $f(x,y)$ is continuous on the Rectangle $R=[a,b]\times [c,d]$

$${\iint }_{R}f(x,y)dA={\int }_{x=a}^{x=b}{\int }_{y=c}^{y=d}f(x,y)dydx={\int }_{y=c}^{y=d}{\int }_{x=a}^{x=b}f(x,y)dxdy$$

These integrals are called iterated integrals. Meaning that there are two options for approaching the integral. One may be easier than the other, so there is strategy involved.

Separable Integrals

If Fubini’s Theorem applies, and $f(x,y)=h(x)\cdot g(y)$

$${\iint }_{R}h(x)g(y)dA=\left({\int }_{x=a}^{x=b}h(x)dx\right)\left({\int }_{y=c}^{y=d}g(y)dy\right)$$