Area Using Integration

by Krutartha
(India)

I have a query regarding calculating the area under a curve...(may be a bit funny to you).

But why is that we only get area when we integrate a given function over definite limits?

How do you prove that the answer that we get finally is the area and not anything else?

Thank you in advance!!

Answer by Pablo:

Here we need to make a clear difference between what is an indefinite integral and a definite integral.

We say that we find an indefinite integral when we perform the inverse operation to differentiation. That is, given a function f(x), we find a function F(x) whose derivative is f(x).

Now, with the definite integral we have something different. We express a definite integral as:

The easiest way to remind you why this thing represents area is the following. Imagine you have the graph of the function. Now, we put the "dx" in there for a reason.

The "dx's" are very little intervals. These are the bases of thin rectangles, and we multiply them by f(x). That's what's inside the integral. This product gives as the area of a rectangle: base times height.

This is well explained in the page: Definite Integrals Made Clear.

Now, the integral sign just means "sum". Well, it is not just any sum, but the limit of a sum. It just means calculate the areas of all thin rectangles and sume them up. You may think about this graph: