Average value over a closed interval | AP Calculus AB | Khan Academy

TL;DR

The average value of a function over a closed interval can be found by dividing the area under the curve by the width of the interval.

Transcript

what I want to do in this video is think about the idea of an average value of a function over some closed interval so what do I mean by that and how could we think about what average value of a function even means so let's say that's my y-axis and let's say that this is my this right over here is my x-axis and let me draw a function here so let's ... Read More

Key Insights

• 😚 The average value of a function over a closed interval represents the average height of the function over that interval.
• 🗂️ Calculating the average value involves determining the area under the curve and dividing it by the width of the interval.
• 😚 The formula for the average value of a function over a closed interval is 1/(B-A) times the definite integral of the function.
• 👻 This formula allows for the calculation of the average value with the knowledge of the definite integral and interval limits.
• ❓ It is important to understand the concept behind the formula rather than just memorizing it.
• 🤔 The average value can be thought of as the average height of the function, which can vary based on the shape of the curve.
• 😫 The average value can also be likened to finding the average of a set of numbers but in a continuous setting.

Questions & Answers

Q: What does "average value of a function over a closed interval" mean?

The average value of a function over a closed interval refers to the average height of the function over that interval.

Q: How can the average value of a function be calculated?

The average value of a function can be calculated by dividing the area under the curve over the interval by the width of the interval.

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

The formula for the average value of a function over a closed interval is 1/(B-A) times the definite integral of the function from A to B.

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

The average value of a function is equal to the area under the curve divided by the width of the interval.

Summary & Key Takeaways

• The average value of a function over a closed interval is the average height of the function over that interval.

• This average height can be found by dividing the area under the curve by the width of the interval.

• The formula for the average value of a function over a closed interval is 1/(B-A) times the definite integral of the function from A to B.