💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

# Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.2) to the function $F(x) = \displaystyle \int_{a}^x f(t) dt$.

## $f(c)(b-a)$

#### Topics

Applications of Integration

### Discussion

You must be signed in to discuss.

Lectures

Join Bootcamp

### Video Transcript

okay, The mean value. Their first off we know efforts continuously closed. Interval A to B and F is also different. Chewable on the open interval. A deadbeat. Therefore, we know there's a number seeing a comma be such that the derivative of prime F C is F A B minus F of a over B minus A. Therefore, we can write this out as half primacy is F A B minus off of a over B minus. And remember the fundamental theme of calculus. It means that we can essentially break up into girls like this. I remember when the integral is from a This means it's just zero, which means we can disregard it. In this case, bring us back to FC is one over B minus eight times the integral from A to B after vax d x. Thus we have now proven the mean value the arm

#### Topics

Applications of Integration

Lectures

Join Bootcamp