Area Problem - Area Problem As noted in the first section...

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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.
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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.
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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.
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