# 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(x)` that is positive on `[a,b]` 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->oo)f(x_i^(**))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^af(x)dx=0`.

Property 2. Inverting bounds of integration: `int_a^bf(x)dx=-int_b^af(x)dx`.

Property 3. If `f` is an even function then `int_(-a)^af(x)dx=2int_0^af(x)dx`.

## 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.