In many situations, the average value of a group of discrete numbers is needed. This is easy to calculate as

      h_ave = (h₁+ h₂+ h₃+ ...... + h_n)/n

But, sometimes the average value of a continuous function f(x) on a domain [a, b] needs to be determined. For example, the average atmospheric temperature over 24 hours.

To determine the average value of a continuous function, first, the domain [a, b] is divided into n equal subintervals, with equal width,

      Δx = (b - a)/n

Average Value of Function
y = 1 + x² on [-1, 2]

Then points x_1^* , x_2^*, x_3^*, ......, x_n^* in each subinterval are used to calculate the average value of each subintervals as f(x_1^*), f(x_2^*), f(x_3^*), ......, f(x_n^*):

Since Δx = (b - a)/n, the above expression becomes

According to the definition of definite integral, the limiting value of the above summation is

Therefore, the average value of a continuous function f(x) on the interval [a, b] is defined as

If function f(x) is continuous on [a, b], then there exist a number c in [a, b] such that

This is called the Mean Value Theorem for Integrals.

The geometric interpretation of the Mean Value Theorem for Integrals is that, for a positive function f(x), there exists a number c between a and b such that the rectangle with base [a, b] and height f(c) has the same area as the region under the graph of f(x) from a to b.