Integrals

So far, we have studied derivatives, which is a method for extracting information about the rate of change of a function. But as you may have realized, integration is the reverse of the earlier problems. 

In integration, we find the area underneath a curve. For example, if we have a car and our function gives us its velocity, the area under the curve will give us the distance it has traveled between two points.

Let's suppose we have the curve , and the area under the curve between x = a (the lower limit) and x = b (the upper limit, also written as [a, b]) is S. Then, we have the following:

The diagramatical representation of the curve is as follows: 

This can also be written as follows:

In the preceding function, the following applies: , and  is in the subinterval .

The function looks like this:

The integral gives us an approximation of the area under the curve such that for some, ε > 0 (ε is assumed to be a small value), the following formula applies:

Now, let's suppose our function lies both above and below the x axis, thus taking on positive and negative values, like so:

As we can see from the preceding screenshot, the portions above the x axis (A₁) have a positive area, and the portions below the x axis (A₂) have a negative area. Therefore, the following formula applies:

Working with sums is an important part of evaluating integrals, and understanding this requires some new rules for sums. Look at the following examples:

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Now, let's explore some of the important properties of integrals, which will help us as we go deeper into the chapter. Look at the following examples:

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Now, suppose we have the function , which looks like this:

Then, we get the following property:

This property only works for functions that are continuous and have adjacent intervals.