Average Value of a Function

Average value of a function on the interval can be approximated by the summation of the value of a function of all subintervals over the number of subintervals. Since definite integration can be considered as the summation of infinitesimal elements on a closed interval. When the number of subintervals approaching infinity, the average value of a function can be determined by the definite integration of the function over its length of integration. Imply

Average Function Value 

For example: function f(x)=(50x)1/3-1

An approximation of function f(x)=(50x)1/3-1 on the interval [-8,6] by dividing the interval into 14 subintervals. The average value of the function f(x) on the interval [-8,6] can be obtained by taking the average of these 14 values.

More accurate average function value can be obtained by increasing the number of subintervals. When the number of subintervals approaching infinity or the length of interval approaching zero, imply

Therefore, the average function value of function  f(x)=(50x)1/3-1 on the closed interval [-8,6] is

Since definite integral is the net value of the sum of infinisimal elements. the average function value is the signed value of the function on the closed interval. Graphically,

Mean Function Value 

Since the function f(x) is continuous on closed interval [a,b], the average funcion value should always lie on the curve of the function, therefore there is alway a number c on the closed interval [a,b] such that mean function value f(c) is equal to the average function value.

The number c can be obtained by equating the average function value and the mean value function.  imply

 Graphically, the intersection of function in an enlarged view is