The First Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus:  Statement and Proof

 The First Fundamental Theorem of Calculus Let f be a continuous function on [a,b] and let            F'(x) = f(x)  then

Proof:

Cut up the interval [a,b] into several pieces with

a  =  x0  <  x1  <  x2  <  x3  <  ...  <  xn-1  <  xn  =  b

Then

F(b) - F(a)  =

[F(xn) - F(xn-1)] + [F(xn-1) - F(xn-2)] +  [F(xn-2) - F(xn-3)] +... + [F(x2) - F(x1)] +  [F(x1) - F(x0)]

=

By the mean value theorem there is a  ci between  xi-1  and  xi with

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

Multiplying both sides by Dxi gives

F'(ci)Dxi  =  F(xi) - F(xi-1)

Substituting into the sum gives

Taking the limit as n approaches infinity, gives the definite integral.

Examples

Example 1:

Example 2:

Find the area bounded by the curve

y = x2 - x , y = 0,  x = 4

Notice that there is area both below the x-axis and above.  We can find: