Mohamed Salahuddin Blog Post 4

Definite integrals and the area under a curve

In this topic I want to explain the Definite integrals, and explain its relation with the area under a curve.

A Definite Integral has start and end values: in other words there is an interval from “a” to “b”.

We can find the Definite Integral by first calculating the Indefinite Integral at points x, or I(x);

The second step is to find substitute the “a” and “b”, to find I(a) and I(b);

The third step is to subtract the I(a) from I(b) to obtain the Definite integrals.

Now, that we know how to solve it, we need to explain what does it mean. Say if it is a Definite integrals of the f(x) from “a” to “b”, it is represented by the area of the curve from the x-axis to the f(x) graphically.

If however f(x) is crossing the x-axis between the interval [a, b], we need to find the crossing point c, and integrated it in two parts, from [a, c] and [c, b].

Now we will use an example to illustrate my point.

Say if we want to integrate f(x) =x-2 from [2, 3].

Graphically, it is exactly the area between the f(x) =x-2 and x-axis from x=2 to x=3.

Now if we want to find the area between [1, 3], we need to be careful, since f(x) is crossing the x-axis within the interval at x=2. So we need to integrate two parts: [1, 2], and [2, 3]

The total area is thus 1+3=4.

Now we can see the close relation between the Definite Integral and areas under a curve. It can also be used to find volumes, central points and many useful things.