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Applications of the Integral
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While there are many integrals that we can calculate explicitly, there are many others that we cannot For example, it is impossible to evaluate e x dx
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That is to say, it can be proved mathematically that no closed-form antiderivative can 2 be written down for the function e x Nevertheless, ( ) is one of the most important integrals in all of mathematics, for it is the Gaussian probability distribution integral that plays such an important role in statistics and probability Thus we need other methods for getting our hands on the value of an integral One method would be to return to the original de nition, that is to the Riemann sums If we need to know the value of
e x dx
then we can approximate this value by a Riemann sum
e x dx e (025) 025 + e (05) 025 + e (075) 025 + e 1 025
2 2 2 2 2
A more accurate approximation could be attained with a ner approximation:
e x dx
10 j =1
e (j 01) 01
( )
CHAPTER 8 Applications of the Integral
e x dx
100 j =1
e (j 001) 001
The trouble with these numerical approximations is that they are calculationally expensive: the degree of accuracy achieved compared to the number of calculations required is not attractive Fortunately, there are more accurate and more rapidly converging methods for calculating integrals with numerical techniques We shall explore some of these in the present section It should be noted, and it is nearly obvious to say so, that the techniques of this section require the use of a computer While the Riemann sum ( ) could be computed by hand with some considerable effort, the Riemann sum ( ) is all but infeasible to do by hand Many times one wishes to approximate an integral by the sum of a thousand terms (if, perhaps, ve decimal places of accuracy are needed) In such an instance, use of a high-speed digital computer is virtually mandatory
THE TRAPEZOID RULE
The method of using Riemann sums to approximate an integral is sometimes called the method of rectangles It is adequate, but it does not converge very quickly and it begs more ef cient methods In this subsection we consider the method of approximating by trapezoids Let f be a continuous function on an interval [a, b] and consider a partition P = {x0 , x1 , , xk } of the interval As usual, we take x0 = a and xk = b We also assume that the partition is uniform
Fig 844
In the method of rectangles we consider a sum of the areas of rectangles Figure 844 shows one rectangle, how it approximates the curve, and what error
Applications of the Integral
is made in this particular approximation The rectangle gives rise to a triangular error region (the difference between the true area under the curve and the area of the rectangle) We put quotation marks around the word triangular since the region in question is not a true triangle but instead is a sort of curvilinear triangle If we instead approximate by trapezoids, as in Fig 845 (which, again, shows just one region), then at least visually the errors seem to be much smaller
Fig 845
In fact, letting x = xj xj 1 as usual, we see that the rst trapezoid in the gure has area [f (x0 )+f (x1 )] x/2 The second has area [f (x1 )+f (x2 )] x/2, and so forth In sum, the aggregate of the areas of all the trapezoids is 1 1 {f (x0 ) + f (x1 )} x + {f (x1 ) + f (x2 )} 2 2 1 + {f (xk 1 ) + f (xk )} x 2 x = {f (x0 ) + 2f (x1 ) + 2f (x2 ) 2 + + 2f (xk 1 ) + f (xk )} x +
It is known that, if the second derivative of f on the interval [a, b] does not exceed M then the approximation given by the sum ( ) is accurate to within M (b a)3 12k 2 [By contrast, the accuracy of the method of rectangles is generally not better than N (b a)2 , 2k where N is an upper bound for the rst derivative of f We see that the method of trapezoids introduces an extra power of (b a) in the numerator
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