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Unformatted text preview: Approximating Definite Integrals In this chapter weve spent quite a bit to time on computing the values of integrals. However, not all integrals can be computed. A perfect example is the following definite integral. We now need to talk a little bit about estimating values of definite integrals. We will look at three different methods, although one should already be familiar to you from your Calculus I days. We will develop all three methods for estimating by thinking of the integral as an area problem and using known shapes to estimate the area under the curve. Lets get first develop the methods and then well try to estimate the integral shown above. Midpoint Rule This is the rule that you should be somewhat familiar to you. We will divide the interval into n subintervals of equal width, We will denote each of the intervals as follows, Then for each interval let be the midpoint of the interval. We then sketch in rectangles for each subinterval with a height of . Here is a graph showing the set up using . We can easily find the area for each of these rectangles and so for a general n we get that, Or, upon factoring out a we get the general Mid Point Rule. Trapezoid Rule For this rule we will do the same set up as for the Midpoint Rule. We will break up the interval into n subintervals of width, Then on each subinterval we will approximate the function with a straight line that is equal to the function values at either endpoint of the interval. Here is a sketch of this case for . Each of these objects is a trapezoid (hence the rules name) and as we can see some...
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