We give more contexts to understand integrals.
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.
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”.
- By how much has the population grown during the first three days of the experiment?
- 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?
- Find the population at any time .
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:
An application of this definition is given in the next example.
What is the average velocity of the object?
(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 .
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.
- 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.