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 Skip Navigation LinksMath Help > Calculus > Calculus Theorems > Fundamental Th. of Calc

Fundamental Theorem of Calculus

If f is the derivative of F, then

ób
ô
õ
a

f(x) dx = F(b) - F(a)

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 < ... < xn-1 < xn = b

where Dxi is the length of the ith subinterval.  If ci 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
õ
a

f(x) dx

Proof of the Fundamental Theorem of Calculus

If f is the derivative of F, then

ób
ô
õ
a

f(x) dx = F(b) - F(a)

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

a = x0 < x1 < x2 < ... < xn-1 < xn = 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 ci) such that

F'(ci) = (F(xi) - F(xi-1)) / (xi - xi-1)

Because F' is f, F'(ci) = f(ci).  We let Dxi = xi - xi-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
õ
a

f(x) dx

 

Related pages in this website

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))/(a-b) for some c in (a,b)

Other so-called "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.

 

The webmaster and author of this Math Help site is Graeme McRae.