We give more contexts to understand integrals.

Velocity and displacement, speed and distance

Some values include ‘‘direction’’ that is relative to some fixed point.

On the other hand speed and distance are values without ‘‘direction.’’

Consider a particle whose velocity at time is given by .
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What is the displacement of the particle from to ? That is, compute: What is the displacement of the particle from to ? That is, compute: What is the distance traveled by the particle from to ? That is, compute: What is the distance traveled by the particle from to ? That is, compute:

Average value

Conceptualizing definite integrals as ‘‘signed area’’ works great as long as one can actually visualize the ‘‘area.’’ In some cases, a better metaphor for integrals comes from the idea of average value. Looking back to your days as an even younger mathematician, you may recall the idea of an average: If we want to know the average value of a function, a naive approach might be to partition the interval into equally spaced subintervals, and choose any in . The average of , , …, is: Multiply this last expression by :

where . Ah! On the right we have a Riemann Sum! Now take the limit as : This leads us to our next definition:

The average value of a function gives the height of a single rectangle whose area is equal to

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An application of this definition is given in the next example.

When we take the average of a finite set of values, it does not matter how we order those values. When we are taking the average value of a function, however, we need to be more careful.

For instance, there are at least two different ways to make sense of a vague phrase like ‘‘The average height of a point on the unit semi circle’’

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One way we can make sense of ‘‘The average height of a point on the unit semi circle’’ is to compute the average value of the function on the interval . Another way we can make sense of ‘‘The average height of a point on the unit semi circle’’ is the average value of the function on , since is the height of the point on the unit circle at the angle .

See if you can understand intuitively why the average using should be larger than the average using .

Mean value theorem for integrals

Just as we have a Mean Value Theorem for Derivatives, we also have a Mean Value Theorem for Integrals.

This is an existential statement. The Mean Value Theorem for Integrals tells us:

The average value of a continuous function is in the range of the function.

We demonstrate the principles involved in this version of the Mean Value Theorem in the following example.