Area Problem

Now It is time to start second kind of integral: Definite Integrals.

 The area problem is to definite integrals what tangent & rate of change problems are to derivatives.

The area problem is one of the interpretations of a definite integral and this will lead us to the definition of definite integral.

To begin we are going to suppose that we've got a function f ( x ) that is positive on some interval [a,b].  What we desire to do is find out the area of the region among the function and the x-axis.

Probably it's easiest to see how we do this with an example.  Therefore let's find out the area between f ( x ) = x² + 1 on [0,2].  In other terms, we desire to find out the area of the shaded region below.

Now, at this instance, we can't do this precisely.  Though, we can estimate the area. We will estimate the area through dividing up the interval in n subintervals each of width,

                                                     Δx = (b - a)/ n

Then in each of interval we can make a rectangle whose height is specified by the function value at a particulate point in the interval.  We can then determine the area of each of these rectangles, add up them and it will be an estimate of the area.