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Area ProblemAs noted in the first section there are two kinds of integrals. To this point in this chapter 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 f(x) 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
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. Let¡¯s find the estimated area. First, the width of each of the rectangles is
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. 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. In the case of a decreasing function 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.
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Acknowledgement: The above material is based on Paul's Online Math Notes. |