LaTex Multiple Integrals Examples $\iint_R 1 \, dA$

 ### Fundamental Double Integral Examples

Here are comprehensive examples of double integrals in LaTeX, drawn from common mathematical contexts such as area, volume, and physics applications. I've included both basic and advanced forms, with iterated integrals, polar coordinates, and special notations like closed surfaces. Each example includes the LaTeX code for rendering.

#### 1. Basic Double Integral Over a Region (Area Calculation)

This computes the area of a region \( R \).

```latex

$$\iint_R 1 \, dA$$

```

#### 2. Volume Under a Surface

Volume under \( z = f(x,y) \) over region \( R \).

```latex

$$\iint_R f(x,y) \, dA$$

```

#### 3. Iterated Double Integral with Rectangular Limits

Integral of \( x + y \) over \( 0 \leq x \leq 1 \), \( 0 \leq y \leq 2 \).

```latex

$$\int_0^1 \int_0^2 (x + y) \, dy \, dx$$

```

#### 4. Double Integral in Polar Coordinates

Over a disk of radius 1.

```latex

$$\int_0^{2\pi} \int_0^1 f(r, \theta) r \, dr \, d\theta$$

```

#### 5. Mass of a Lamina with Density Function

Mass \( m \) with density \( \delta(x,y) \).

```latex

$$m = \iint_R \delta(x,y) \, dA$$

```

#### 6. Center of Mass (x-coordinate)

For a lamina.

```latex

$$\bar{x} = \frac{1}{m} \iint_R x \delta(x,y) \, dA$$

```

#### 7. Double Integral with Region Subscript

Placing the region \( R \) under the integral sign using limits.

```latex

$$\iint\limits_R f(x,y) \, dA$$

```

#### 8. Closed Surface Double Integral (Flux)

For vector fields, often in Gauss's theorem.

```latex

\oiint_S \mathbf{F} \cdot d\mathbf{S}

```

(Note: Define `\oiint` as a custom command, e.g., `\newcommand{\oiint}{\iint\!\!\!\!\!\!{\scriptstyle \subset\!\supset}}` for a closed symbol.)

#### 9. Double Integral of a Specific Function Over a Triangle

Over the region \( 0 \leq y \leq x \), \( 0 \leq x \leq 1 \).

```latex

$$\int_0^1 \int_0^x e^{y/x} \, dy \, dx$$

```

#### 10. Average Value Over a Region

Average of \( f(x,y) \) over \( R \).

```latex

$$\frac{1}{\iint_R 1 \, dA} \iint_R f(x,y) \, dA$$

```

#### 11. Probability Density Double Integral

Joint probability over region \( R \).

```latex

$$P = \iint_R p(x,y) \, dx \, dy$$

```

#### 12. Moment of Inertia About x-Axis

For a lamina.

```latex

$$I_x = \iint_R y^2 \delta(x,y) \, dA$$

```

#### 13. Double Integral in Cylindrical Coordinates

Volume integral.

```latex

$$\int_0^{2\pi} \int_0^a \int_0^{f(r,\theta)} r \, dz \, dr \, d\theta$$

```

(Note: This is effectively a triple integral, but the inner is single, forming a double base.)

#### 14. Surface Integral Over a Graph

Parametrized form.

```latex

$$\iint_D g(u,v) \left\| \mathbf{r}_u \times \mathbf{r}_v \right\| \, du \, dv$$

```

#### 15. Electric Flux Through a Surface

Using Gauss's law.

```latex

\Phi_E = \oiint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q}{\epsilon_0}

```

These examples cover a wide range of applications. To render them, use a LaTeX compiler like Overleaf, enclosing in math mode (e.g., `\[ ... \]` for display). For packages, include `\usepackage{amsmath}` for symbols like `\iint`. If you need solutions or evaluations for specific ones, let me know!