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calc2p377 - 12 Use the Midpoint Rule with the given value...

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Unformatted text preview: 12 Use the Midpoint Rule with the given value of n to approx- te the integral. Round the answer to four decimal places. { Llox/x3 + ldx, n = 4 I0. Jew/2 cos 4xdx, n = 4 rsin(x2)dx, n = 5 12. fxz e’de, n = 4 p 0 ‘ If you have a CAS that evaluates midpoint approximations and graphs the corresponding rectangles (use middlesum , and middlebox commands in Maple), check the answer to 5 Exercise 11 and illustrate with a graph. Then repeat with n= lOandn= 20. VWith a programmable calculator or computer (see the instruc— tions for Exefcise 7 in Section 5.1), compute the left and right Riemann sums for the function f (x) = sin(x2) on the interval [0, 1] with n = 100. Explain why these estimates show that 0.306 < J; sin(x2) dx < 0.315 .Deduce that the approximation using the Midpoint Rule with n = 5 in Exercise 11 is accurate to two decimal places. Use a calculator or computer to make a table of values of right Riemann sums R for the integral {87 sin x dx with n = 5, 10,50 and 100. What value do these numbers appear 0 be approaching? ’ se a calculator or computer to make a table of values of left and right Riemann sums L, and R for the integral .0 "dx with n = 5 10,50, and 100. Between what two F-oumbers must the value of the integral lie? Can you make a imilar statement for the integral file Fdx? Explain. 0 Express the limit as a definite integral on the given a1. 71 E x,- 111(1 + x1?) Ax? [2! 6] i=1 —m COS X, Ax, [77, 277] fi./:x;k + (xi-EV Ax, [1,8] =1 n E[4 °° i=1 - 306?“)2 + 6(X?‘)5] Ax, [0, 7-] //’7 SECTION 5.2 THE DEFINITE INTEGRAL h. 26. (a) Find an approximation to the integral I: (x2 — 3x) dx using a Riemann sum with right endpoints and n = 8. (b) Draw a diagram like Figure 3 to illustrate the approxi- mation in part (a). (c) Use Theorem 4 to evaluate f; (x2 — 3x) dx. (d) Interpret the integral in part (c) as a difference of areas and illustrate with a diagram like Figure 4. b2_aZ 27. Prove that Jlb x dx = 2 [as—a3 28. Prove that I!) x2 dx = 3 29—30 Express the integral as a limit of Riemann sums. Do not evaluate the limit. 6 x 10 29. f2 1+x5 30. L (x — 4lnx)dx dx (AS '~ Use the form of the definition of the integral given in Lrn 4 to evaluate the integral. l(1+3x)dx 22. J14(x2+2x—5)dx 5 (2 — x2)dx 24. f0 (1 + 2x3) dx x3 dx 3 l—32 Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. 3|. foflsinSxdx 32. Lloxd xd6 33. The graph ’of f is shown. Evaluate each integral by inter- preting it in terms of areas. (a) flit-001x (b) from (c) jgmdx (d) {from ya 2 y=f(x) O 2 4 6 8 ) 34. The graph of 9 consists of two straight lines and a semicircle. Use it to evaluate each integral. (b) f: 90:) dx (a) j: 90:) dx (c) j; 900 dx ...
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