Mean Value Theorems for Integrals, Proof, Example

Mean value theorem defines that a continuous function has at least one point where the function equals its average value.

Mean Value Theorems for Integrals | Integration

Statement
    This theorem states that the slope of a line merging any two points on a 'smooth' curve will be the same as the slope of the line tangent to the curve at a point between the two points.
   
Let f be the continuous function on [a, b]. Then the average or mean f(c) of c on [ a,b ] is
    Mean value theorem for integrals

Model Diagram
   
mean value theorem for integrals example


Proof
    As f is continuous on [a, b], f has minimum value m and maximum value M on [a, b], and
Mean value theorem for integration example

Then divide the above equation by (b - a) which gives,
Mean value theorem

If f is a constant then choose as m = a and M = b, so that m < M
As f is continuous on [m,M] and Mean value theorem lies between f(m) and f(M), by the intermediate value theorem there exists c in [m,M], thus in [a,b], such that:
Mean value theorem for integration proof

Hence the Mean Value Theorems for Integrals / Integration is proved.


Example
    Find the average value of f(x)=7x2 - 2x - 3 on the interval [2,6].

Solution
    In the given equation f is continuous on [2, 6].


    Here 44 denotes the average value of the given function.
    Now substitute x=16 in the given original function.
    f(x)=7x2 - 2x2 - 3 = 44
    =7x2 - 2x2 - 47
    =(x + 2.45)(x - 2.74)=0
   
X=-2.45 , 2.74 Of these two values, only 2.74, is in given interval, hence 2.74 is the desired value of c.
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