Average Value of a Function

The average value of a function on an interval is the constant height that would produce the same total area as the original curve over that interval. The integral gives the accumulated area under f(x), and dividing by the interval length b-a converts that total into an area-per-unit-length, which is exactly the mean height. This works because a continuous function can be thought of as many thin rectangles whose average height is the total area spread evenly across the interval.

Use this formula when you need the mean output of a quantity that changes continuously with x, such as temperature over a day, velocity over a time period, or a rate like concentration or intensity across a region. It is especially useful in physics, engineering, and calculus problems involving “average height” or “average rate” over an interval.

• Forgetting to divide by the interval length b-a after integrating
• Using the endpoints themselves as the average instead of integrating the function over the whole interval
• Computing the antiderivative correctly but evaluating the definite integral with the limits in the wrong order