The First Mean Value Theorem for Riemann Integration states that if $f(x)$ is a continuous function on the closed interval $[a, b]$, then there exists a point $c$ in the interval $[a, b]$ such that:

$f(c) \cdot (b - a) = \int_a^b f(x) \, dx$

Explanation:

1. Function Continuity: The theorem requires that $f(x)$ be continuous on the interval $[a, b]$.
2. Existence of $c$: The theorem guarantees the existence of a point $c \in [a, b]$ such that the value of the function at this point, multiplied by the length of the interval $(b - a)$, is equal to the area under the curve from $a$ to $b$ (i.e., the definite integral of $f(x)$ over $[a, b]$).
3. Interpretation: Geometrically, this means that there is at least one point on the curve $y = f(x)$ where the rectangle with height $f(c)$ and width $(b - a)$ has the same area as the area under the curve between $x = a$ and $x = b$.

Summary:

• $f(c)$ represents the average value of the function over the interval $[a, b]$.
• The point $c$ is the "mean" point where the function $f(x)$ equals this average value over the interval.

Would you like more details on any part of this theorem or its applications?

Here are 5 related questions for further exploration:

1. How does the First Mean Value Theorem differ from the Second Mean Value Theorem for Riemann Integration?
2. Can the First Mean Value Theorem be extended to functions that are not continuous?
3. How can the First Mean Value Theorem be applied in calculating average values of functions in real-world scenarios?
4. What is the relationship between the First Mean Value Theorem and the Fundamental Theorem of Calculus?
5. How does the First Mean Value Theorem relate to the Mean Value Theorem for differentiation?

Tip: When applying the First Mean Value Theorem, remember that the existence of the point $c$ is guaranteed by the continuity of $f(x)$. The theorem does not provide a method for finding $c$, but it assures that such a point exists.