The average value of a function represents the mean height of the function over a specified interval. This concept is fundamental in calculus applications and connects to the Mean Value Theorem for Integrals, providing insights into function behavior and area calculations.

Geometric Interpretation

Geometrically, the average value represents the height of a rectangle with width (b-a) that has the same area as the area under the curve y = f(x) from x = a to x = b. This rectangle has area equal to the definite integral, and its height is the average value. It's like finding the "average height" of the function.

• Area under curve = ∫ₐᵇ f(x) dx
• Rectangle area = height × width = average value × (b-a)
• Setting them equal: average value × (b-a) = ∫ₐᵇ f(x) dx
• Therefore: average value = (1/(b-a)) × ∫ₐᵇ f(x) dx

Sources & References