What is fundamental about the Fundamental Theorem of Calculus?

Answer 1

Calculus consists of two main topics -- differentiation and integration. Differentiation is concerned with 'rates of change', for example the rate at which the position of a moving object is changing with respect to time, otherwise known as velocity. Integration is concerning with computing areas,volumes, and lenght by first approximating the region as the sum of many smaller regions which are simpler to compute, and taking the limit as the number of smaller regions increases to infinity.

At first sight, it doesn't seem like these two topics -- differentiation and integration -- have anything to do with one another. In fact, in a calculus course either one could be presented first since it wouldn't require knowledge of the other one (traditionally, differentiation is taught first, then integration, but it isn't necessary to do them in this order.)

However, the amazing fact is that these two seemingly unrelated problems are completely intertwined. The Fundamental Theorem of Calculus, one of the most amazing and profound results in all of mathematics, spells out just how the processes of differentiation and integration are related -- they are essentially reverse operations of one another, or two sides of the same coin. The Fundamental Theorem of Calculus is so named because it ties together the two main themes of the subject. F(x) = f (t)dt is a function then we have = f(t)dt but since Since dt = h we have - f (x) = f (t) - f(x)dt . and using the continuity of F(t), we have the following equality. - f (x) = 0 .Now, the punch line!The function F(x) is differentiable and F '(x) = f (x).

Many calculus books have two parts to the FTC (Fundamental Theorem of Calculus)

Part one states that the area under a section of a curve is the antiderivative evaulated at the upper limit minus the lower limit. That is:

Integral ( f(x) dx) from a to b = F(b) - F(a)

where b is the upper boundary and a is the lower boundary

and

Part two states that the derivate of integration is the integrand:

d/dx integral (f(t) dt) from 0 to x = f(x)

where x is the upper boundary and 0 is the lower boundary.

So what went into the integral that you derive is the result.

Note: it really helps to see the pictures of what is going on.