Let f (x) be continuous on [a, b]. Set
As the name "First Mean Value Theorem" seems to imply, there is also a Second Mean Value Theorem for Integrals:
Second Mean Value Theorem for Integrals. Let f (x) and
g(x) be continuous on [a, b]. Assume that g(x) is positive,
i.e.
g(x) 0 for any
x
[a, b]. Then there exists
c
(a, b) such that
As an application one may define the Center of Mass of one-dimensional non-homogeneous objects such as a metal rod. If the object is homogeneous and lying on the x-axis from x = a to x = b, then its center of mass is simply the midpoint
Example. A rod of length L is placed on the x-axis from
x = 0 to x = L. Assume that the density
(x) of the rod
is proportional to the distance from the x = 0 endpoint of the
rod. Let us find the total mass M and the center of mass xc
of the rod. We have
(x) = kx, for some constant k > 0. We have
Exercise 1. Find the average value of
Exercise 2. Find the average value of
Exercise 3. Assume that f (x) be continuous and increasing
on [a, b]. Compare
[0,
/4].
[0, 4].
f (x)dx .
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