Define the average value of an integrable function of two variables

The average value of an integrable function \( f(x, y) \) of two variables over a region \( R \) in the \( xy \)-plane is defined as the integral of the function over the region \( R \) divided by the area of \( R \). Mathematically, this is expressed as:

\[
\text{Average value of } f(x, y) \text{ over } R = \frac{1}{\text{Area of } R} \iint_R f(x, y) \, dA
\]

Here's a step-by-step breakdown of how this definition is applied:

1. **Determine the Region \( R \)**:
   Identify the region \( R \) in the \( xy \)-plane over which the average value is to be calculated. This region could be rectangular, circular, or any other shape.

2. **Calculate the Area of \( R \)**:
   Find the area of the region \( R \), denoted as \( \text{Area of } R \). For simple geometric shapes, the area can often be computed directly. For more complex shapes, it might involve integration:
   \[
   \text{Area of } R = \iint_R 1 \, dA
   \]

3. **Set Up the Double Integral**:
   Set up the double integral of \( f(x, y) \) over the region \( R \):
   \[
   \iint_R f(x, y) \, dA
   \]

4. **Compute the Double Integral**:
   Evaluate the double integral to find the total value of the function over the region \( R \).

5. **Divide by the Area**:
   Divide the result of the double integral by the area of the region \( R \):
   \[
   \text{Average value of } f(x, y) \text{ over } R = \frac{1}{\text{Area of } R} \iint_R f(x, y) \, dA
   \]

**Example:**

Let's say \( f(x, y) = x + y \) and we want to find its average value over the rectangular region \( R \) defined by \( 0 \leq x \leq 2 \) and \( 0 \leq y \leq 3 \).

1. **Determine the Region \( R \)**:
   The region \( R \) is a rectangle with \( 0 \leq x \leq 2 \) and \( 0 \leq y \leq 3 \).

2. **Calculate the Area of \( R \)**:
   \[
   \text{Area of } R = \int_0^2 \int_0^3 1 \, dy \, dx = 2 \cdot 3 = 6
   \]

3. **Set Up the Double Integral**:
   \[
   \iint_R (x + y) \, dA = \int_0^2 \int_0^3 (x + y) \, dy \, dx
   \]

4. **Compute the Double Integral**:
   \[
   \int_0^2 \left[ \int_0^3 (x + y) \, dy \right] dx = \int_0^2 \left[ xy + \frac{y^2}{2} \right]_0^3 dx = \int_0^2 \left( 3x + \frac{9}{2} \right) dx
   \]
   \[
   = \int_0^2 \left( 3x + 4.5 \right) dx = \left[ \frac{3x^2}{2} + 4.5x \right]_0^2 = \left( \frac{3(2)^2}{2} + 4.5 \cdot 2 \right) - (0 + 0) = 6 + 9 = 15
   \]

5. **Divide by the Area**:
   \[
   \text{Average value of } f(x, y) \text{ over } R = \frac{1}{6} \iint_R (x + y) \, dA = \frac{15}{6} = 2.5
   \]

Therefore, the average value of \( f(x, y) = x + y \) over the region \( R \) is \( 2.5 \).

Do you want details or have any questions?

Here are some related questions you might be interested in:

1. How do you evaluate double integrals over non-rectangular regions?
2. What is the difference between the average value of a function over a region and over an interval?
3. How can you apply the concept of average value to functions of three variables?
4. What are some practical applications of finding the average value of a function?
5. How do you handle changing the order of integration in double integrals?
6. Can the average value of a function be used to approximate integrals?
7. How do you find the average value of a periodic function?
8. What are the steps to convert a double integral from Cartesian to polar coordinates?

**Tip:** When calculating the average value of a function over a region, ensure that you properly set up the limits of integration to match the region \( R \).

Learn how to define and calculate the average value of an integrable function of two variables over a given region in the xy-plane. Includes step-by-step examples and applications. Suitable for advanced undergraduate students and those interested in mathematical analysis.

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