# FIND THE VALUES OF C THAT SATISFY THE MEAN VALUE THEOREM FOR INTEGRALS

If f is continuous on [a, b], then at some point c in [a, b], The graphical rectangular interpretation of the Mean value theorem for Definite Integrals is that:

If f is continuous on [a, b], then at some point c in [a, b] there is a rectangle with height f(c), and length b – a, such as the area of the rectangle equals the area under the curve f(x) on the interval [a, b]

For each problem, find the values of c that satisfy the Mean Value Theorem for Integrals.

Problem 1 :

f(x) = x + 2; [-3, 2]

Solution :

Finding average value of the function :

Here a = -3, b = 2

Finding the value of c that lies in the given interval :

f(x) = x + 2

f(c) = c + 2

1.5 = c + 2

c = 1.5 - 2

c =  -0.5

So, the value of c is -0.5, which lies in the interval.

Problem 2 :

Solution :

Finding average value of the function :

Here a = 3, b = 5

Equating the value derived from mean value theorem for integrals to f(x), we get

So, the value of c is (9 + 2√3)/3, which lies in the interval.

Problem 3 :

Solution :

Finding average value of the function :

Here a = 2, b = 3

Equating the value derived from mean value theorem for integrals to f(x), we get

So, the value of c is √6, which lies in the interval.

Problem 4 :

Solution :

Finding average value of the function :

Here a = 0, b = 3

Equating the value derived from mean value theorem for integrals to f(x), we get

So, the value of c is 4/3, which lies in the interval.

Problem 5 :

Solution :

Finding average value of the function :

Here a = 1, b = 3

Equating the value derived from mean value theorem for integrals to f(x), we get

So, the value of c is -1 + 2√2 , which lies in the interval.

Problem 6 :

Solution :

Finding average value of the function :

Here a = 1, b = 4

Equating the value derived from mean value theorem for integrals to f(x), we get

So, the value of c is 7/3 , which lies in the interval.

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