# Category: Definite Integral

## Area Problem Revisited

We have already talked about the area problem and have presented one approach to solve this problem. Here, we are going to demonstrate a different approach.

So, suppose that we are given a function ${f{{\left({x}\right)}}}$ that is positive on ${\left[{a},{b}\right]}$ and we want to find the area ${S}$ under the curve. It is not so easy to find the area of a region with curved sides. So, we will start from rectangle approximation.

## Concept of Definite Integral

In the Area Problem note, we saw that the limit of the form $\lim_{{{n}\to\infty}}{f{{\left({{x}_{{i}}^{{\star}}}\right)}}}\Delta{x}$ arises when we compute an area.

It turns out that the same type of limit occurs in a wide variety of situations even when ${f{}}$ is not necessarily a positive function.

## Properties of Definite Integrals

Now let's see what properties integral has.

Property 1. It follows from the definition of integral that ${\int_{{a}}^{{a}}}{f{{\left({x}\right)}}}{d}{x}={0}$.

Property 2. Inverting bounds of integration: ${\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{d}{x}=-{\int_{{b}}^{{a}}}{f{{\left({x}\right)}}}{d}{x}$.

## The Fundamental Theorem of Calculus

When we introduced definite integrals, we computed them according to the definition as the limit of Riemann sums and we saw that this procedure is not very easy. In fact, there is a much simpler method for evaluating integrals.