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.