91Ó°ÊÓ

When we talk about a definite integral, we're looking at calculating the area under a curve within a specified range. This range is denoted from \(a\) to \(b\). A definite integral can be written as \( \int_{a}^{b} f(x) \, dx \). This expression symbolizes the accumulation of all the small areas (rectangles) under the function \(f(x)\) within this interval.

Key characteristics include:

• The limits \(a\) and \(b\) specify the starting and ending points on the function's x-axis.
• The outcome of a definite integral is a number, which represents the total area.

The definite integral is fundamental in finding the average value of a function over a given interval. It translates to asking, "What is the appropriate measure of this function’s overall behavior between \(a\) and \(b\)?" With this concept, problems like the one in our exercise find clarity as it ensures the link between constants and variable functions over specific intervals.

A constant function is one of the simplest mathematical constructs: it doesn't change no matter what x-value you plug into it. Mathematically, it's expressed as \( f(x) = c \), where \(c\) is a fixed number.

In the context of definite integrals, the integral of a constant function over an interval \([a, b]\) can be easily calculated. The formula shows that it's simply the product of the constant \(c\) and the length of the interval \((b-a)\). This is written as:

• \( \int_{a}^{b} c \, dx = c(b-a) \)

Applying this to our exercise, if the average value \( \text{av}(f) \) of a function over \([a, b]\) behaves as a constant function, its integral should ideally give us the same result as the integral of the function \(f(x)\) itself over the same interval.

An integrable function is one that meets the criteria for integration over a specified interval. This means it can be assigned a definite integral in the range \([a, b]\). In simpler words, these functions have a finite area embodied under their curves over these intervals.

Main characteristics include:

• The function must be well-behaved, meaning it should not exhibit undefined behavior like discontinuities across the interval.
• Its definite integral \( \int_{a}^{b} f(x) \, dx \) produces a finite result.

Recognizing a function as integrable is crucial before solving problems related to average values via definite integrals. The term highlights its capability to relate averages to a sum, which, when divided by the interval length \(b-a\), gives us the constant \( \text{av}(f) \) capable of representing the function throughout that range. Ensuring a function is integrable allows us to apply the concept of definite integrals meaningfully in practical calculus problems.