Calculating average value of function over interval | AP Calculus AB | Khan Academy

TL;DR

The average value of a function on a closed interval can be found by taking the definite integral of the function over the interval and dividing it by the width of the interval.

Transcript

• Let's say that we have the function F of X is equal to X squared plus one and what we want to do is we want to figure out the average value of our function F on the interval, on the closed interval between zero and let's say between zero and three. I encourage you to pause this video especially if you've seen the other videos on introducing the i... Read More

Key Insights

• 🗂️ Calculating the average value of a function on an interval involves finding the area under the curve and dividing it by the width of the interval.
• ❓ The average value can be different from the maximum or minimum values of the function on the interval.
• 😥 The mean value theorem for integrals states that there exists a point in the interval where the function takes on its average value.

Questions & Answers

Q: What is the formula for finding the average value of a function on a closed interval?

The average value of a function on a closed interval is equal to the definite integral of the function over the interval, divided by the width of the interval.

Q: How is the average value related to the area under the curve?

The average value of a function represents the height of a rectangle that has the same area as the region under the curve over the interval.

Q: Can the average value of a function be lower or higher than its maximum or minimum values on the interval?

Yes, the average value can be different from the maximum or minimum values. It represents the balance point of the region under the curve and may not coincide with extremal points.

Q: Is there a connection between the average value of a function and the mean value theorem for integrals?

Yes, the mean value theorem for integrals states that there exists at least one point in the interval where the function takes on its average value. This is consistent with the idea that the average height multiplied by the width gives the same area as the region under the curve.

Summary & Key Takeaways

• The function F(x) = x^2 + 1 is used as an example to find the average value on the closed interval between zero and three.

• The graph of the function is plotted, and the area under the curve represents the average value.

• The definite integral of F(x) over the interval is calculated and divided by the width of the interval to find the average value, which in this case is four.