Area Problem
As noted in the first section of this section there are two
kinds of integrals and to this point we�ve looked at indefinite integrals. It
is now time to start thinking about the second kind of integral : Definite
Integrals. However, before we do that we�re going to take a look at the Area
Problem. The area problem is to definite integrals what the tangent and rate of
change problems are to derivatives.
The area problem will give us one of the interpretations of
a definite integral and it will lead us to the definition of the definite
integral.
To start off we are going to assume that we�ve got a
function
 that
is positive on some interval [a,b]. What we want to do is determine the
area of the region between the function and the x-axis.
It�s probably easiest to see how we do this with an
example. So let�s determine the area between
 on
[0,2]. In other words, we want to determine the area of the shaded region
below.
Now, at this point, we can�t do this exactly. However, we
can estimate the area. We will estimate the area by dividing up the interval
into n subintervals each of width,
Then in each interval we can form a rectangle whose height
is given by the function value at a specific point in the interval. We can then
find the area of each of these rectangles, add them up and this will be an
estimate of the area.
It�s probably easier to see this with a sketch of the
situation. So, let�s divide up the interval into 4 subintervals and use the
function value at the right endpoint of each interval to define the height of
the rectangle. This gives,
Note that by choosing the height as we did each of the
rectangles will over estimate the area since each rectangle takes in more area
than the graph each time. Now let�s estimate the area. First, the width of
each of the rectangles is
 .
The height of each rectangle is determined by the function value at the right
endpoint and so the height of each rectangle is nothing more that the function
value at the right endpoint. Here is the estimated area.
Of course taking the rectangle heights to be the function
value at the right endpoint is not our only option. We could have taken the
rectangle heights to be the function value at the left endpoint. Using the left
endpoints as the heights of the rectangles will give the following graph and
estimated area.
In this case we can see that the estimation will be an
underestimation since each rectangle misses some of the area each time.
There is one more common point for getting the heights of
the rectangles that is often more accurate. Instead of using the right or left
endpoints of each sub interval we could take the midpoint of each subinterval as
the height of each rectangle. Here is the graph for this case.
So, it looks like each rectangle will over and under
estimate the area. This means that the approximation this time should be much
better than the previous two choices of points. Here is the estimation for this
case.
We�ve now got three estimates. For comparison�s sake the
exact area is
So, both the right and left endpoint estimation did not do
all that great of a job at the estimation. The midpoint estimation however did
quite well.
Be careful to not draw any conclusion about how choosing
each of the points will affect our estimation. In this case, because we are
working with an increasing function choosing the right endpoints will
overestimate and choosing left endpoint will underestimate.
If we were to work with a decreasing function we would get
the opposite results. For decreasing functions the right endpoints will
underestimate and the left endpoints will overestimate.
Also, if we had a function that both increased and
decreased in the interval we would, in all likelihood, not even be able to
determine if we would get an overestimation or underestimation.
Now, let�s suppose that we want a better estimation,
because none of the estimations above really did all that great of a job at
estimating the area. We could try to find a different point to use for the
height of each rectangle but that would be cumbersome and there wouldn�t be any
guarantee that the estimation would in fact be better. Also, we would like a
method for getting better approximations that would work for any function we
would chose to work with and if we just pick new points that may not work for
other functions.
The easiest way to get a better approximation is to take
more rectangles (i.e. increase n). Let�s double the number of
rectangles that we used and see what happens. Here are the graphs showing the
eight rectangles and the estimations for each of the three choices for rectangle
heights that we used above.
Here are the area estimations for each of these cases.
So, increasing the number of rectangles did improve the
accuracy of the estimation as we�d guessed that it would
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