# Double integrals over a general region

### Double integrals over a general region

#### Lessons

Notes:

Double integrals of a multi-variable function give the volume under the function $f(x,y)$

Double Integrals Over General Regions

Suppose the region is not rectangular, but rather the region is between two curves. For example, we want to integrate within the following region $D$, where $D$ is:

$a \leq x \leq b$
$g_1 (x) \leq y \leq g_2(x)$

Then the iterated integral will be:

$\int \int_D f(x,y)dA = \int_a^b \int_{g_1(x)}^{g_2(x)}f(x,y)dydx$

Likewise, suppose we have the region $D$, where $D$ is:

$g_1 (y) \leq x \leq g_2(y)$
$a \leq y \leq b$

Then the iterated integral will be:

$\int \int_D f(x,y)dA = \int_a^b \int_{g_1(y)}^{g_2(y)} f(x,y)dxdy$

You usually find the region $D$ yourself.

Properties of Double Integrals

The three properties of double integrals are the following:

$\int \int_D f(x,y) + g(x,y) dA = \int\int_D f(x,y)dA \int \int_D g(x,y) dA$
$\int \int_D cf(x,y) dA = c \int \int_D f(x,y) dA, \mathrm{where\;c\; is\;a\;constant}$
$\int \int_D f(x,y) = \int \int_{D_1} f(x,y) + \int \int_{D_2} f(x,y) \;\mathrm{where\; D\; is\; split \; into\; 2\; regions\; D_1 \; \& \; D_2}$

Volume of General Regions in 3D

Suppose you want to find the volume of a region that is above $g(x,y)$ and below $f(x,y)$, bounded by a region $D$. Then, the volume is:

$\int \int_D f(x,y) - g(x,y)dA$

• Introduction
Double Integrals Over a General Region Overview:
a)
Double Integrals Over General Regions
• Not a rectangular region
• Region is between two curves
• Case 1: Two curves in terms of $x$
• Case 2: Two curves in terms of $y$
• An Example

b)
Properties of Double Integrals
• 3 properties
• Sum of two functions
• Factoring the scalar
• Splitting $D$ into 2 regions: $D_1$ & $D_2$
• An Example of using the properties

c)
Volume of General Region in 3D
• Subtract the two functions
• Find the region $D$
• An Example

• 1.
Evaluating the Double Integral Over General Regions
Evaluate $\int \int_{D} x - y^{2} dA, \,$ where $D$ is the region bounded by $\, y = \frac{1}{2}x \,$ and $\, y = \sqrt{x}$.

• 2.
Evaluate $\int \int_{D} \frac{1}{y} \, dA, \,$ where $D$ is the region bounded by $\, x = -y \,$ and $\, x = y^{2}$.

• 3.
Evaluate $\int \int_{D} xy - x \, dA, \,$ where $D$ is the region bounded by $\, y = x^{2} - 1 \,$ and $\, x-$axis.

• 4.
Evaluate $\int \int_{D} e^{1-x} \, dA, \,$ where $D$ is the region bounded by the triangle with vertices (0,0), (2, 0), (0, 2).

• 5.
Find the area of the region bounded by $\, y = 1 - x^{2} \,$ and $\, y = x^{2} - 1$.