
If f is the derivative of F, then  ó^{b} 
f(x) dx = F(b)  F(a) 
Before we prove the Fundamental Theorem of Calculus, let's define a few terms...
(Georg Friedrich Bernhard Riemann (18261866) was most famous for work in nonEuclidean geometry, differential equations, and number theory. His results in physics and mathematics form the basis of Einstein's theory of general relativity.)
Let f be defined on [a,b], and let D be a partition of [a,b] given by
a = x_{0} < x_{1} < x_{2} < ... < x_{n1} < x_{n} = b
where Dx_{i} is the length of the ith subinterval. If c_{i} is any point in the ith subinterval then the sum
_{n }S
^{i=1}f(c_{i}) Dx_{i}, x_{i1} <= c_{i} <= x_{i}
is called a Riemann sum of f for the partition D.
The limit as the length of the largest subinterval of partition D (the norm of the partition, denoted IIDII) approaches zero (if it exists) is the definite integral, denoted
ó^{b}
ô
õ_{a}f(x) dx
If f is the derivative of F, then  ó^{b} 
f(x) dx = F(b)  F(a) 
Let D be a partition of [a,b] with
a = x_{0} < x_{1} < x_{2} < ... < x_{n1} < x_{n} = b
Using this partition, F(b)F(a) can be rewritten as
_{n }S
^{i=1}( F(x_{i})  F(x_{i1}) )
By the Mean Value Theorem, there exists a number in each subinterval (call it c_{i}) such that
F'(c_{i}) = (F(x_{i})  F(x_{i1})) / (x_{i}  x_{i1})
Because F' is f, F'(c_{i}) = f(c_{i}). We let Dx_{i} = x_{i}  x_{i1} , which means we can rewrite the sum, above, as
F(b)  F(a) = _{n }S
^{i=1}f(c_{i}) Dx_{i}
Taking the limit as IIDII —> 0,
F(b)  F(a) = ó^{b}
ô
õ_{a}f(x) dx
Definition of Integral as the limit as Dx goes to zero of a Riemann Sum
Mean Value Theorem  that f'(c) = (f(a)f(b))/(ab) for some c in (a,b)
Other socalled "fundamental" theorems
The Fundamental Theorem of Arithmetic says every number has exactly one unique prime factorization.
The Fundamental Theorem of Algebra says Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.
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