14. Definite integrals and area

14.1Introduction¶

As we mentioned at the beginning of the previous chapter, the main problem we are dealing with in this part of the course is regarding the area - how to find the area under the graph of a function?

Intuitively, the area under the graph of the function $f(x)$ can be found by approximating it rectangles.

Now that we have some intuitive understanding of how to compute the area, we want to know how can this be done mathematically. We have already done the first step in the previous chapter in which we learned how to compute indefinite integrals. The rest of the answer was also outlined in the previous chapter:

• learn how to find definite integrals

• use definite integrals to compute the area

14.2Definite integrals¶

In this section we are going to learn how to find definite integral of a given function $f(x)$, which will be denoted as $\displaystyle \int _a ^b f(x) \, dx$, where $a,b$ are just some numbers. The following formula tells us how to actually find definite integrals.

Therefore, we see that indeed we first need to know how to compute an indefinite integral of a function in order to be able to find its definite integral.

14.3Area¶

In this section we are going to use the definite integral to compute the area. Namely, the area under the graph of the function $f(x)$ over the segment $[a,b]$ is equal to the value of the definite integral $\displaystyle \int_a^b f(x) \, dx$.