Fundamental Theorem of Calculus Proof

The fundamental theorem of calculus determines the association the two basic operations of calculus called as the differentiation and integration. The first fundamental theorem of integration deals with the indefinite integration and the second theorem of integration deal with the definite integral of the function.

In this article we are going to see about the proof of the fundamental theorem for calculus.

Proof of fundamental theorem for calculus:

Proof of first fundamental theorem:

Let us take a real valued function f which is given by

x
F(x) = int f(t) dt, here f is also a real valued function
a

Then F is said to be continuous on [a,b] and can be differentiated on the open interval (a,b) which is given by

F′(x) = f(x), for all the values of x in the interval (a,b)

Let us consider two numbers x1 and x1+∆x , in the closed interval (a, b), then we have

x1                                                                        x1+∆x
F(x1) = int f(t) dt   -------> (1)     and       F(x1+∆x) = int f(t) dt ----------> (2)
a                                                            a

When we subtract the first equation and the second equation we get

x1+∆x                x1
F(x1+∆x) - F(x1) =  int   f(t) dt  -  int   f(t) dt ------------> (a)
a                    a

a              x1+∆x
=  int f(t) dt + int f(t) dt
x1                 a

a              x1+∆x           x1+∆x
int f(t) dt + int f(t) dt =  int f(t) dt
x1                 a                x1

When we substitute in equation (a) we get,

x1+∆x
F(x1+∆x) - F(x1) =  int   f(t) dt  -------------> (b)
x1

According to the theorem of integration

x1+∆x
int   f(t) dt  = f(c) ∆x  ,  given that there exists c in [x1 and x1+∆x]
x1

When we substitute this value in equation (b) we get,

F(x1+∆x) - F(x1) = f(c) ∆x

When we divide by ∆x on both sides we get,

F(x1+∆x) - F(x1) = f(c)
∆x

Take the limit ∆x  ->0, on both sides we get,

lim      F(x1+∆x) - F(x1) = lim  f(c)
∆x ->0            ∆x                        ∆x ->0

This left side of the equation is the derivative of F at x1

F′(x1)  = lim  f(c) -----> (3)
∆x ->0

We know that

lim    x1 = x1  and lim    x1+∆x  = x1
∆x ->0                 x1+∆x->0

Then according to the squeeze theorem, we get,

lim    c  = x1
∆x ->0

Substituting this in (c), we get

F′(x1) = lim  f(c)
c -> x1

The function f is continuous and real valued and thus we get,

F′(x1) = f(x1)

Hence the proof of the fundamental theorem for calculus.

Corollary of fundamental theorem proof:

The fundamental theorem is used to calculate the definite integral of a given function f if f is a real-valued continuous function on the interval [a, b], then

b
int f(x) dx = F(b) – F(a)