Math Problem Statement

Second mean value theorem

Solution

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

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

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

Key Points:

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

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

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Integral Calculus

Formulas

-

Theorems

Second Mean Value Theorem for Integrals

Suitable Grade Level

Advanced Undergraduate