StellarThe average value of a function on an interval [a,b] is the total accumulated value over the interval divided by the interval’s length. The formula is . You can think of this as the continuous version of an arithmetic mean: instead of averaging a list of numbers, you average all function values across an interval. Geometrically, the average value is the height of a horizontal line that would create a rectangle with the same area as the area under the curve on that interval.
This concept is useful because it turns a varying function into one representative constant value over an interval. If a velocity function is averaged, you get average velocity over the interval. If a temperature function is averaged, you get the average temperature across that time period. The key difference between average value of a function and average rate of change is that average value uses an integral and measures typical output height, while average rate of change measures overall change in output divided by change in input.