We give more contexts to understand integrals.

Velocity and displacement, speed and distance

Some values include “direction” that is relative to some fixed point.

  • is the velocity of an object at time . This represents the “change in position” at time .
  • is the position of an object at time . This gives location with respect to the origin.
  • is the displacement, the distance between the starting and finishing locations.

On the other hand speed and distance are values without “direction.”

  • is the speed.
  • is the distance traveled.
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:

Change in the amount

We can apply the Fundamental Theorems of Calculus to a variety of problems where both accumulation and rate of change play important roles. For example, we can consider a tank that is being filled with fuel at some rate. Given the rate, we can ask what is the amount of fuel in the tank at a certain time. Or, a tank that is being emptied at a given rate, or a culture of bacteria growing in a Petri dish, or a population of a city, etc. This brings us to our next theorem.

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 on the interval , a naive approach might be to introduce equally spaced grid points on the interval and choose a sample point in each interval , .

We will approximate the average value of on the interval with the average of , , …, and : Multiply this last expression by :

where . Ah! On the right we have a Riemann Sum!

What will happen as ?

We take the limit as : This leads us to our next definition:

Multiplying this equation by , we obtain that If is positive, 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.

Choose all the correct expressions for , the average velocity of an object moving along a straight line over the time interval .

(Reminder: is the position function, and the acceleration).

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.