L4-integrationAppsAvgValue.mws
Calculus II
Lesson 4: Applications of Integration 2: Average Value of a Function
Let's examine why we should define the
average value
of a function
over the interval
to be
We could start by partitioning the interval
into
equal sub-intervals. Since
should be approximately constant on each sub-interval (at least if
is large), we could approximate the average value of
by computing the average value of the function
we get by replacing
on each subinterval by its value at the right-hand endpoint of the sub-interval. This is illustrated below, taking
to be the function
on
, with 6 sub-intervals. (The graph of the function
consists of the 6 line segments that make up the tops of the rectangles in the figure.)
>
student[rightbox](cos(x), x=0..Pi/2,6);
The average value of
should just be the mean of the
values taken by
. (
in the example above.) This mean is
=
. Apart from the first factor of
, we recognise this as a Riemann sum for
; as
, our approximation should therefore converge to
, so it seems reasonable to define the (exact) average value of
to be this last expression.
Example
If a freely falling body starts from rest, then its displacement is given by
. Fix a time
, and let the velocity after time
be V.
(a). Show that the average velocity, with respect to
, over the interval
, is
.
Solution.
First, notice that we are asked to find the average value of
velocity
,
. Since
is the velocity at time
,
. The average velocity, with respect to time, over the interval
, is
>
restart;
>
(1/(T - 0))*int(g*t, t=0..T);
and this is indeed
.
(b). Define
, the displacement at time
. Write the velocity as a function of displacement,
, and show that the average velocity, with respect to s, over the interval
, is
.
Solution.
In part (a), all quantities were written as functions of
. Now we have to write everything in terms of
, through the relation
. Solving for
, we get
, so (for example)
=
. Now let's compute the average of
, considered as a function of
, over the interval
:
>
S := g*T^2/2;
>
w := (1/(S - 0))*int(sqrt(2*g*s), s=0..S);
>
assume(g>0); assume(T>0);
>
simplify(w);