Fundamental Theorem of Calculus

Before we prove the Fundamental Theorem of Calculus, let's define a few terms...

Riemann Sum

(Georg Friedrich Bernhard Riemann (1826-1866) was most famous for work in non-Euclidean 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 = x₀ < x₁ < x₂ < ... < x_n-1 < x_n = b

where Dx_i is the length of the ith subinterval.  If c_i is any point in the ith sub-interval then the sum

n S i=1

f(ci) Dxi,   xi-1 <= ci <= xi

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

ób ô õa

f(x) dx

Proof of the Fundamental Theorem of Calculus

Let D be a partition of [a,b] with

a = x₀ < x₁ < x₂ < ... < x_n-1 < x_n = b

Using this partition, F(b)-F(a) can be rewritten as

n S i=1

( F(xi) - F(xi-1) )

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_i-1)) / (x_i - x_i-1)

Because F' is f, F'(c_i) = f(c_i).  We let Dx_i = x_i - x_i-1 , which means we can rewrite the sum, above, as

F(b) - F(a) =

n S i=1

f(ci) Dxi

Taking the limit as IIDII --> 0,

F(b) - F(a) =

ób ô õa

f(x) dx