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Keyuan Liu Assignment Assignment9 due 03/04/2016 at 10:59pm PST 4. (1 pt) Improper Integrals, Innite Integrands: Compute the improper integrals...

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Keyuan Liu Math3B-01-W16-Paul Assignment Assignment9 due 03/04/2016 at 10:59pm PST 1. (1 pt) Determine whether the integral is divergent or con- vergent. If it is convergent, evaluate it. If not, give the answer -1. Z 4 xe - 3 x dx Answer(s) submitted: (incorrect) 2. (1 pt) Find what value of c does R 5 c x 4 dx = 1 ? Answer: Answer(s) submitted: (incorrect) 3. (1 pt) Improper Integrals, Infinite Limits of Integra- tion: Integrals may have infinite limits of integration, or inte- grands that have singularities. Such integrals are called ”im- proper” even though there is nothing wrong with such integrals. They may have well defined values, in which case we say they converge, or they may not, in which case we say they diverge. In this and the next problem, give the value of the integral if it converges, and enter the letter ”D” if it diverges. R 1 1 x 2 d x = . R 1 1 x d x = . R 1 e - x d x = . R 1 e x d x = . R - 1 1 + x 2 d x = . R π - e x d x = . Answer(s) submitted: (incorrect) 4. (1 pt) Improper Integrals, Infinite Integrands: Com- pute the improper integrals below. Enter the letter ”D” if they diverge. R 1 0 1 x 2 d x = . R 1 0 1 x d x = . R 1 0 1 x d x = . R 1 0 ln x d x = . R 1 - 1 1 1 - x 2 d x = . Answer(s) submitted: (incorrect) 5. (1 pt) Determine whether the integral is divergent or con- vergent. If it is convergent, evaluate it. If not, state your answer as ”divergent”. Z - (( x + 3 ) 2 - 6 ) dx Answer(s) submitted: (incorrect) 6. (1 pt) The improper integral Z - xdx is A. convergent since the area to the left of x = 0 cancels with the area to the right of x = 0. B. convergent since it equals lim a →- Z 0 a xdx + lim b Z b 0 xdx = - + = 0. C. divergent since both integrals Z 0 - xdx = - and Z 0 xdx = + are divergent. D. divergent by comparison to Z - xe - x dx . 1
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E. convergent since it equals lim t Z t - t xdx = lim t ± t 2 2 - ( - t ) 2 2 ² = 0. F. divergent since Z 0 - xdx is convergent and Z 0 xdx is divergent. G. divergent by comparison to Z - xdx . Answer(s) submitted: (incorrect) 7. (1 pt) Determine if the improper integral converges and, if so, eval- uate it. R 2 dx x - 1 . A. 0 B. Diverges C. 1 D. 2 Answer(s) submitted: (incorrect) 8. (1 pt) Determine if the improper integral converges and, if so, eval- uate it. R 0 dx 6 + x = Write F if the integral doesn’t converge. Answer(s) submitted: (incorrect) 9. (1 pt) Determine whether the integral is divergent or con- vergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as “infinity”. If it diverges to negative infinity, state your answer as “-infinity”. If it diverges without being infinity or negative infinity, state your answer as “divergent”. Z 8 1 8 3 x - 1 dx = . Answer(s) submitted: (incorrect) 10. (1 pt) The integral Z 0 3 x ( 1 + x ) dx is improper for two reasons: the interval [ 0 , ] is infinite and the integrand has an infinite discontinuity at x = 0. Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: Z 0 3 x ( 1 + x ) dx = Z 1 0 3 x ( 1 + x ) dx + Z 1 3 x ( 1 + x ) dx If the improper integral diverges, type an upper-case ”D”. Answer(s) submitted: (incorrect) 11. (1 pt) Find the value of the constant C for which the integral Z 0 ± x x 2 + 1 - C 3 x + 1 ² dx converges. Evaluate the integral for this value of C . C = Value of convergent integral = Answer(s) submitted: (incorrect) 12. (1 pt) Consider the integral Z 1 0 - 7ln ( x ) x dx If the integral is divergent, type an upper-case ”D”. Otherwise, evaluate the integral. Answer(s) submitted: (incorrect) 13. (1 pt) Consider the integral Z 0 6cos 2 ( α ) d α If the integral is divergent, type an upper-case ”D”. Otherwise, evaluate the integral. Answer(s) submitted: (incorrect) 14. (1 pt) Consider the integral Z π 0 - 9sec ( x ) dx If the integral is divergent, type an upper-case ”D”. Otherwise, evaluate the integral. Answer(s) submitted: 2
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