{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

RandolphT2W98 - improper integral converges or diverges If...

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

View Full Document Right Arrow Icon
Winter 1992 Math 21 Test #2 Name 1. (10 points each) Find only FIVE of the following six indefinite integrals. Show each step. Circle the integral for problem that you DO NOT want to be graded . ( This will be graded on an all-or-nothing basis for 4 points extra credit.) (a) Z cos(3 x ) cos(4 x ) dx (b) Z x - 3 x 2 + 1 dx (c) Z 1 4 x 2 + 1 dx (d) Z tan 3 x dx (e) Z x 2 ln x dx (f) Z p 9 - x 2 dx 2. (10) Use partial fractions to express the integral as a sum of simpler integrals. Z 5 x 2 + 3 x - 2 x 2 ( x + 2) dx DO NOT EVALUATE the integral. 3. (a) (4 pts) Use the trapezoid rule with n = 4 to approximate Z 1 0 e - x 2 dx . (b) (6 pts) If T 4 is the number you found in (a), how large is the difference: fl fl fl fl Z 1 0 e - x 2 dx - T 4 fl fl fl fl ? You may use the fact that for f ( x ) = e - x 2 , f 00 ( x ) 2 for x [0 , 1]. (c) (6 pts) How large should n be in order for the trapezoid rule to produce a value, T n , that approxi- mates R 1 0 e - x 2 dx to within 0.001? 4. (6 pts) A definite integral has been calculated for you below. Determine whether the corresponding
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: improper integral converges or diverges . If it converges, specify its value. Z t 1 ln x x 2 dx = ( using “parts” ) =-ln t t-1 t + 1. Therefore, Z ∞ 1 ln x x 2 dx = 5. (10) Show all your work in determining the value (if it exists) of the following improper integral. If it diverges show your work in determining this. Z 3 2 3 √ x-2 dx 6. (8) Use the comparison thoerem, explaining how it applies, to determine whether the integral is convergent or divergent . You DO NOT need to evaluate the integral. (You may use, without proof, any knowledge about convergence or divergence of the improper integrals involving 1 /x , 1 /x 2 , e-x , etc., but be specific.) Z ∞ 1 sin 2 x x 2 dx...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern