The **Second Mean Value Theorem for Integrals** (sometimes called the Mean Value Theorem for Definite Integrals) states the following:

If \( f \) is continuous on the closed interval \([a, b]\) and \( g \) is integrable and of constant sign (either always non-negative or always non-positive) on \([a, b]\), then there exists a point \( c \in [a, b] \) such that:

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

### Key Points:

1. The function \( f(x) \) must be continuous on \([a, b]\).
2. The function \( g(x) \) must be integrable on \([a, b]\) and have a constant sign (it doesn’t change from positive to negative or vice versa).
3. The point \( c \) is within the closed interval \([a, b]\).

This theorem provides a weighted average interpretation for integrals, stating that the integral of \( f(x)g(x) \) over \([a, b]\) is the same as \( f \) evaluated at some point \( c \), multiplied by the integral of \( g(x) \) over \([a, b]\).

Would you like more details or have any questions?

Here are some related questions:

1. How does the Second Mean Value Theorem differ from the First Mean Value Theorem?
2. What is the geometric interpretation of the Second Mean Value Theorem?
3. Can the Second Mean Value Theorem be applied if \( g(x) \) changes sign?
4. How is the Second Mean Value Theorem used in physics or engineering?
5. What are some examples where the Second Mean Value Theorem is useful?

**Tip:** The Second Mean Value Theorem is particularly useful in integral approximations and in situations involving non-uniform distributions, where \( g(x) \) acts as a weighting function.

Explore the Second Mean Value Theorem for Integrals, a key concept in calculus. Learn how this theorem relates to integrals of continuous functions and their weighted averages. Suitable for advanced undergraduate students and those interested in integral calculus.

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