{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

calc2p377

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern