Properties of the Definite Integral: Intuition

Why do we need to learn these properties of the definite integral? These properties will lead to the fundamental theorem of calculus. This theorem is arguably one of the most useful in all of mathematics. It transforms the integral from a mathematical curiosity to a powerful tool that is used in science, engineering, economics and many other fields.

And what are these properties? You will find that most of these you've already seen before. Limits and indefinite integrals had similar properties. After all, definite integrals are just limits.

There are other properties, though, that are a little more special. They are related to the notion of area...


Properties of the Definite Integral: Property 1

The first property is that you can take constants out of the integral sign. This is just like what we do with limits. After all, integrals are limits.

In mathematical symbols, this is:

first property of integrals

As an example:

first property of integrals


Properties of the Definite Integral: Property 2

The integral of the sum of two functions equals the sum of the integrals of each function. This is just like limits and indefinite integrals. For example:

first property of integrals


Properties of the Definite Integral: Property 3

This property is called the integral mean value theorem. It has a simple geometric interpretation. Let's say we have a function f(x):

simple function on an interval

Let's also take two points a and b. We learned in definite integrals that the integral simple function on an interval is the area under the graph of f(x) from a to b. That is, it is this area:

simple function on an interval

Let's think a bit about this area. This geometric shape has base of length (b-a). For example, if we had a=2 and b=6, this shape would had base of length 4.

Now, let's take a rectangle with that same base:

rectangle

What property 3 says is that there exists a rectangle with that base that has area equal to the integral. Moreover, that rectangle has height equal to the value of the function at some point in the interval.

All that means is that these two areas are equal:

rectangle

This concept is pretty intuitive. There must be a rectangle with that base that has area equal to the integral.

Writing this in precise symbols, the area under that rectangle is base times height:

area of rectangle

Where x sub zero is some point in the x axis between a and b. And as we said, this area equals the integral:

mean value theorem for integrals

And this is a pretty cool equality.


Properties of the Definite Integral: Property 4

Let's say we have a function f(x), and three points on the x axis: a, b and c:

decomposition of area

We have that a < b < c. The definite integral

integral

gives us the area under the curve from a to c:

integral

The integral from a to b equals this area:

integral

And the integral from b to c equals this area:

integral

We can see clearly from the graphs that if we sum these two last areas we'll get the area from a to c. That is:

integral

And that is exactly what property 4 says. Moreover, these numbers don't need to be in the order I put them. That is, it is not necessary for them to hold the relationship: a < b < c.


Conclusion

These are the most important properties of definite integrals. The first two are just like the properties of limits. The other two are very intuitive and relate to the concept of area.

You just need to remember that integrals represent areas and you'll be fine.

And this is an exciting moment! If you understood these properties, you're ready to digest the fundamental theorem of calculus...

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