4. Double Integrals Over General Regions

Type I and Type II Regions

Definition (Type I)

A region of type I (or $y$-simple region) is a region of the form

$D=\{(x,y)|a\le x\le b,g_1(x)\le y\le g_2(x)\}$

Definition (Type II)

A region of type II (or $x$-simple region) is a region of the form

$D=\{(x,y)|c\le y\le d,h_1(y)\le x\le h_2(y)\}$

Integral over Type I Region

Theorem

If $f(x,y)$ is a function on a type I region $D$ such that

$D=\{(x,y)|a\le x\le b,g_1(x)\le y\le g_2(x)\}$

then

${\iint }_{D}f(x,y)dA={\int }_a^b{\int }_{g_1(x)}^{g_2(x)}f(x,y)dydx$

Theorem

If $f(x,y)$ is a function on a type II region $D$ such that

$D=\{(x,y)|c\le y\le d,h_1(x)\le x\le h_2(x)\}$

then

${\iint }_{D}f(x,y)dA={\int }_c^d{\int }_{h_1(y)}^{h_2(y)}f(x,y)dxdy$

Steps

How to calculate the double integral ${\iint }_{D}f(x,y)dA$?

1. Sketch the domain $D$.
2. Determine whether it is a type I or II region.
3. Write $D$ in the relevant form (e.g. here Type I):

$D=(x,y)|a\le x\le b,g_1(x)\le y\le g_2(x)$

5. Write the integral as an iterated integral and calculate it

${\iint }_{D}f(x,y)dA={\int }_a^b{\int }_{g_1(x)}^{g_2(x)}f(x,y)dydx=\dots$