Average Value of A Function Calculator Double Integral

What is the Average Value of a Function?

The average value of a function over a region provides a single value that represents the "typical" value of the function within that region. For functions of two variables, this is calculated using a double integral.

Mathematically, the average value is defined as the integral of the function over the region divided by the area of the region. This gives a weighted average where points where the function is higher contribute more to the average.

Double Integral Formula

The average value f̄ of a function f(x,y) over a rectangular region R defined by a ≤ x ≤ b and c ≤ y ≤ d is given by:

Where:

• f(x,y) is the function to evaluate
• R is the region of integration
• dA is the differential area element
• (b - a)(d - c) is the area of the rectangular region

How to Calculate the Average Value

1. Define the function f(x,y) you want to evaluate
2. Specify the rectangular region R with bounds a, b, c, d
3. Calculate the double integral of f(x,y) over R
4. Divide the result by the area of the region (b - a)(d - c)

Example Calculation

Let's find the average value of f(x,y) = x² + y² over the rectangle R = [0,2] × [0,3].

1. First compute the double integral:
   ∫∫_R (x² + y²) dA = ∫₀³ ∫₀² (x² + y²) dx dy
2. Evaluate the inner integral with respect to x:
   ∫₀² (x² + y²) dx = [x³/3 + x y²]₀² = (8/3 + 2y²) - (0 + 0) = 8/3 + 2y²
3. Now evaluate the outer integral with respect to y:
   ∫₀³ (8/3 + 2y²) dy = [8y/3 + 2y³/3]₀³ = (8 + 18) - (0 + 0) = 26
4. Calculate the area of the region:
   Area = (2 - 0)(3 - 0) = 6
5. Finally, compute the average value:
   f̄ = 26 / 6 ≈ 4.333