Newton-Leibniz theorem

Let be such function that the (continuous) function is its derivative i.e or is the primitive function of then the definite integral is the area under the curve drawn by (positive) and

ProofEdit

We will limit ourselves to the positive function in the interval  . Let us estimate the area under the graph of the function   by dividing densely the interval   into sub-intervals with the ending points   and with the length   and such that   and  . If the   is small then between the two consecutive nodes   and   the function   is approximately linear and therefore the area under it can be approximated by the trapezoid area

 

Then the total area under the graph of the function will be estimated as

 

Let   be the primitive function of the function  . Then we can also estimate its derivative as

 

Now in the formula for   we can re-express each value of   by values of   and   (the same   in nominator and denominator cancels out)

 

Because the majority of the sum contributions show up it the sum twice but with the opposite sign and therefore cancels out we get the expression for the total area with the accuracy up the the small term in  ,   which contains the endpoints terms, as

 .

This proves the theorem. Note that the prove readily extends to the function   that is nether continuous or positive everywhere when the above arguing is used piecewise in intervals between the sign change (when the function   is negative the area between the x-axis and the function graph is than taken negative or with the negative sign) or between the discontinuity (the primitive function   stays continuous but its curve may have sharp edges and   may have a sudden jump in value).