Fundamental Theorem(s) of Calculus


The area shaded in red stripes can be estimated as h times ƒ(x). Alternatively, if the function A(x) were known, it could be computed as A(x + h) − A(x). These two values are approximately equal, particularly for small h.

Computing the derivative of a function and “finding the area” under its curve are "opposite" operations
More exactly, the theorem deals with definite integration with variable upper limit and arbitrarily selected lower limit. This particular kind of definite integration allows us to compute one of the infinitely many antiderivatives of a function (except for those which do not have a zero). Hence, it is almost equivalent to indefinite integration, defined by most authors as an operation which yields any one of the possible antiderivatives of a function, including those without a zero. This is a good starting point. Yes, it's all new but he provides the big picture, "the Forest," so to speak, not just the trees. Gets you acquainted with:

Functions
Limits
Derivatives
Antiderivatives / Indefinite Integrals
Definite Integrals

Now you're talking!

Googles
WA
2nd Theorem Interactive Applet

Fundamental theorem of calculus in multiple dimensions:

In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the famous integral theorems of vector calculus:

Gradient theorem
Stokes' theorem
Divergence theorem
Green's theorem

In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.

The good prof states that FTC1 is the more important of the two, and is used most often.
If, F'(x) = f(x), i.e, f(x) is the derivative of F(x)
then, the definite integral of f(x) over [a,b] = F(b) - F(a), i.e, the antiderivatives at the end points of f(x) Example 1

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