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.

### 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.

Let denote the amount of some substance/population at the time .

Assume that the function is continuous, differentiable and its derivative, , continuous on some time interval .

Then, the **change** in the amount over the time interval is given by

The right hand side of the equation gives the “accumulation of a rate”. You can think of this definite integral as the limit of Riemann sums, where each Riemann sum is the“sum of the changes of the amount over small intervals of time”.

- (a)
- By how much has the population grown during the first three days of the experiment?
- (b)
- Compute the right Riemann sum of the function and the interval with . What does this Riemann sum approximate? Is this approximation an underestimate or an overestimate and why?
- (c)
- Find the population at any time .

### 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 toAn application of this definition is given in the next example.

What is the average velocity of the object?

**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”

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.**

- Proof
- Define an accumulation (area) function, , Since is continuous on the interval and differentiable on the interval , we can apply the Mean Value Theorem to the function on the interval . Therefore, there exist a number in such that But we know that , and that . Therefore,

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