By A.R. Gourlay

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Find the exact area under the cosine curve y cos x from x 0 to x b, where 0 ഛ b ഛ ͞2. ) In particular, what is the area if b ͞2? 2 25. Let A be the area under the graph of an increasing contin- Rn Ϫ L n 28. Find the exact area of the region under the graph of y eϪx from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b). the curve y x 3 from 0 to 1 as a limit. (b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E.

5 y (3, 2) y=x-1 A¡ 0 A™ 1 3 x _1 FIGURE 10 The Midpoint Rule We often choose the sample point x*i to be the right endpoint of the i th subinterval because it is convenient for computing the limit. But if the purpose is to find an approximation to an integral, it is usually better to choose x*i to be the midpoint of the interval, which we denote by x i . Any Riemann sum is an approximation to an integral, but if we use midpoints we get the following approximation. 7 shows how the Midpoint Rule estimates improve as n increases.

NOTE 1 The symbol x was introduced by Leibniz and is called an integral sign. It is an elongated S and was chosen because an integral is a limit of sums. In the notation xab f ͑x͒ dx, f ͑x͒ is called the integrand and a and b are called the limits of integration; a is the lower limit and b is the upper limit. For now, the symbol dx has no meaning by itself; xab f ͑x͒ dx is all one symbol. The dx simply indicates that the independent variable is x. The procedure of calculating an integral is called integration.