3. Double Integrals Over Rectangles

From Double Integral to Iterated Integral

Suppose $R$ is the rectangle $[a,b]\times [c,d]$ and $f(x,y)$ is a function.

${\iint }_{R}f(x,y)dA={\sum }_{i=1}^{m}Volume(slicei)$

$\approx {\sum }_{i=1}^{m}Area(slicei)△x$

$={\sum }_{i=1}^m\left({\int }_c^{d}f(x_i^{\ast },y)dy\right)△x$

Fubini’s Theorem

Theorem

If $f$ is continuous on the rectangle

$R=[a,b]\times [c,d]=(x,y)|a\le x\le b,c\le y\le d,$

then

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