The **First Mean Value Theorem** (also known as the Mean Value Theorem for Integrals) is an important result in calculus. It provides a way to relate the integral of a function over an interval to the value of the function at a specific point within that interval. 

### Statement of the First Mean Value Theorem

Let \( f(x) \) be a continuous function on the closed interval \([a, b]\). Then, there exists a point \( c \) in the interval \((a, b)\) such that:

\[
\int_{a}^{b} f(x) \, dx = f(c) \cdot (b - a)
\]

### Intuitive Explanation

The theorem states that for a continuous function on an interval, the area under the curve (the integral) is equal to the area of a rectangle with width \( (b - a) \) and height \( f(c) \) for some \( c \) in the interval \((a, b)\).

### Proof of the First Mean Value Theorem

1. **Define the Function \( F(x) \):**

   Define a function \( F(x) \) as the integral of \( f(t) \) from \( a \) to \( x \):

   \[
   F(x) = \int_{a}^{x} f(t) \, dt
   \]

   Since \( f(x) \) is continuous on \([a, b]\), \( F(x) \) is also continuous on \([a, b]\) and differentiable on \((a, b)\).

2. **Apply the Fundamental Theorem of Calculus:**

   According to the Fundamental Theorem of Calculus, the derivative of \( F(x) \) with respect to \( x \) is \( f(x) \):

   \[
   F'(x) = f(x)
   \]

3. **Apply the Mean Value Theorem:**

   The Mean Value Theorem (MVT) for derivatives states that if \( F(x) \) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists a point \( c \) in \((a, b)\) such that:

   \[
   F(b) - F(a) = F'(c) \cdot (b - a)
   \]

   Substituting \( F'(c) = f(c) \):

   \[
   \int_{a}^{b} f(x) \, dx = F(b) - F(a) = f(c) \cdot (b - a)
   \]

   Here, \( F(b) = \int_{a}^{b} f(x) \, dx \) and \( F(a) = 0 \) since \( \int_{a}^{a} f(x) \, dx = 0 \).

4. **Conclusion:**

   Thus, there exists a point \( c \) in \( (a, b) \) such that:

   \[
   \int_{a}^{b} f(x) \, dx = f(c) \cdot (b - a)
   \]

This completes the proof of the First Mean Value Theorem.

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Would you like more details or have any questions about the theorem? Here are some related topics you might find interesting:

1. Second Mean Value Theorem for Integrals
2. Mean Value Theorem for Derivatives
3. Applications of the First Mean Value Theorem
4. Geometric Interpretation of the Mean Value Theorem
5. Difference between the Mean Value Theorem and the Intermediate Value Theorem

**Tip:** When applying the First Mean Value Theorem, remember that it requires the function to be continuous on the closed interval \([a, b]\). This ensures that the function attains every value between its extremes.

Explore the First Mean Value Theorem in calculus with a detailed proof and intuitive explanation. Learn how this theorem relates the integral of a function over an interval to the value of the function at a specific point within that interval. Suitable for advanced college-level students and calculus enthusiasts.

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