Expert: Ahmed Salami - 8/16/2010

Question
sir,
how a definite integral represents area of the function between the limits and the bounded axis. if there is a function say F(x) whose derivative is f(x). then if we integrate f(x) from a to b, how we are getting its area. dont give me answer that we are dividing the area into infinite number of parts and summing as others. i think you
understand my question more clearly by reading the two paragraphs below.

the relation between f(x) and F(x) is the derivative. but a change in y coordinates of y=F(x) is representing the area between the curve f(x) and one of axes from those those corresponding coordinates of x.
how is this possible.
how do we explore the relation between f(x) and F(x)( which is derivative) as the relation between coordinates of F(x) and area formed by f(x).

  i know by integrating we are doing the antiderivative of f(x). i know how the riemann's summation gives the area of a function. but how by integrating we are getting this area. in the interpretation of derivative we show clearly how it represents slope of the tangent of F(x) at a point (i.e. , we show it clearly as the limit of  change in
y with respect to change in x as change in x tends to 0). but in the interpretation of integral in all text books i saw it is like: "riemanns summation gives area. it is represented by |a to b f(x)dx (where | a to b represents definite integral symbol from a to b) and by integrating(finding anti derivative) and substituting the limits we are finding the area. my question is but how finding anti derivative is equivalent to dividing the given area into infinite no of parts.

thank you sir.


Answer
Hi Venky,
I think I understand what you're pointing at. But you need to understand that calculus became a major part of modern mathematics because some of its manipulations have surprisingly useful consequences, most of which can be non-intuitive.
What I would say to you is that you might not see how it works out intuitively for now, but the beauty of calculus lies in the consistency of its results with real life and natural solutions. Also, try to learn more about limits and i'm sure that very soon you'll start to get a good feel for all of this, trust me. Just always remember that its not always INTUITIVE.

Regards