Chapter 5 | Integration
527
48. [T] Use a computer algebra system to compute the Riemann sum, L N , for N =10, 30, 50 for f ( x ) = sin 2 x on [0, 2 π ]. Compare these estimates with π . In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R N and L N for N = 1,10,100. How do these estimates compare with the exact answers, which you can find via geometry? 49. [T] y =cos( πx ) on the interval [0, 1] 50. [T] y =3 x +2 on the interval ⎡ ⎣ 3, 5 ⎤ ⎦ In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R N and L N for N = 1,10,100. 51. [T] y = x 4 −5 x 2 +4 on the interval [−2, 2], which has an exact area of 32 15 52. [T] y = ln x on the interval [1, 2], which has an exact area of 2ln(2)−1 53. Explain why, if f ( a ) ≥0 and f is increasing on ⎡ ⎣ a , b ⎤ ⎦ , that the left endpoint estimate is a lower bound for the area below the graph of f on ⎡ ⎣ a , b ⎤ ⎦ . 54. Explain why, if f ( b ) ≥0 and f is decreasing on ⎡ ⎣ a , b ⎤ ⎦ , that the left endpoint estimate is an upper bound for the area below the graph of f on ⎡ ⎣ a , b ⎤ ⎦ .
57. For each of the three graphs: a. Obtain a lower bound L ( A ) for the area enclosed by the curve by adding the areas of the squares enclosed completely by the curve. b. Obtain an upper bound U ( A ) for the area by adding to L ( A ) the areas B ( A ) of the squares enclosed partially by the curve.
55.
Show that,
in
general,
R N − L N = ( b − a )× f ( b )− f ( a ) N . 56. Explain why, if f is increasing on ⎡ ⎣ a , b ⎤
⎦ , the error between either L N or R N and the area A below the graph of f is at most ( b − a ) f ( b )− f ( a ) N .
58. In the previous exercise, explain why L ( A ) gets no smaller while U ( A ) gets no larger as the squares are subdivided into four boxes of equal area.
Made with FlippingBook - professional solution for displaying marketing and sales documents online