Area problem, Mathematics

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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.

587_Area Problem.png

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.


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