# Category: Definite Integral

## Area Problem Revisited

We already talked about area problem and presented there one approach to solve this problem. Here we will present another approach.

So, suppose we are given a function $$${f{{\left({x}\right)}}}$$$ that is positive on $$${\left[{a},{b}\right]}$$$ and we want to find an 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 Area Problem note we saw that 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 this 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 definition as a 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.