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onegirl Group Title

Determine whether the integral converges or diverges. Find the value of the integral if it converges.

  • one year ago
  • one year ago

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  1. onegirl Group Title
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    • one year ago
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  2. onegirl Group Title
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    @Reaper534 can u help?

    • one year ago
  3. terenzreignz Group Title
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    Let's just focus on the integral first, keeping in mind that... \[\Large \int\limits_1^\infty x^{-\frac43}= \lim_{b\rightarrow\infty}\int\limits_1^bx^{-\frac43}\]

    • one year ago
  4. onegirl Group Title
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    okay

    • one year ago
  5. terenzreignz Group Title
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    Well, can you integrate \[\Large \int x ^{-\frac43}=\color{red}?\]

    • one year ago
  6. onegirl Group Title
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    yes i get -3/3sqrt(x)

    • one year ago
  7. terenzreignz Group Title
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    This \[\Large \frac{-3}{3\sqrt{x}}\]?

    • one year ago
  8. onegirl Group Title
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    yes!

    • one year ago
  9. terenzreignz Group Title
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    Or this \[\Large \frac{-3}{\sqrt[3]x}\]

    • one year ago
  10. onegirl Group Title
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    the second one

    • one year ago
  11. terenzreignz Group Title
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    lol... it's cube root, not 3sqrt because \(\Large3\sqrt x\) means something entirely different, ok?

    • one year ago
  12. onegirl Group Title
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    Okay

    • one year ago
  13. terenzreignz Group Title
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    Okay, so, we have... \[\Large \int\limits_1^b x^{-\frac43}=\left.\frac{-3}{\sqrt[3]x}\right]_1^b\]

    • one year ago
  14. terenzreignz Group Title
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    Can you evaluate this bit?

    • one year ago
  15. onegirl Group Title
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    i think

    • one year ago
  16. terenzreignz Group Title
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    Well then, what do you get?

    • one year ago
  17. onegirl Group Title
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    hmm i'm working on it

    • one year ago
  18. onegirl Group Title
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    eh...i'm getting a wrong answer :/

    • one year ago
  19. terenzreignz Group Title
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    Fundamental theorem of Calculus? \[\Large \int\limits_a^b f'(x)dx = \left.f(x)\right]_a^b = f(b)-f(a)\]

    • one year ago
  20. onegirl Group Title
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    ohhhh okay

    • one year ago
  21. terenzreignz Group Title
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    So... \[\Large \int\limits_1^b x^{-\frac43} \ dx=\left.\frac{-3}{\sqrt[3]x}\right]_1^b=\color{red}?\]

    • one year ago
  22. onegirl Group Title
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    so i plug in b and 1 into that then subtract?

    • one year ago
  23. terenzreignz Group Title
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    Yes... but that's not the end yet, just plug in for now, and tell me what you get :)

    • one year ago
  24. onegirl Group Title
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    okay so -3/3sqrt(b) - -3/3sqrt(1)

    • one year ago
  25. terenzreignz Group Title
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    I'm assuming by 3sqrt you mean cube root :D Okay, that being the case, you're right :) \[\Large \int\limits_1^b x^{-\frac43}=\frac{-3}{\sqrt[3]b}+\frac3{\sqrt[3]1}= -\frac3{\sqrt[3]b}+3\] Catch me so far?

    • one year ago
  26. onegirl Group Title
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    yes

    • one year ago
  27. terenzreignz Group Title
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    Now, we're supposed to take the improper integral to infinity, right? Remember this... \[\Large \int\limits_1^\infty x^{-\frac43} \ dx = \color{red}{\lim_{b\rightarrow\infty}}\int\limits_1^bx^{-\frac43} \ dx\] Now is the time to apply that limit (which we haven't done yet) \[\Large \color{red}{\lim_{b\rightarrow\infty}}\left(-\frac3{\sqrt[3]b}+3\right) \]

    • one year ago
  28. onegirl Group Title
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    okay

    • one year ago
  29. terenzreignz Group Title
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    So... evaluating the limit...?

    • one year ago
  30. onegirl Group Title
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    umm i get 3? :/

    • one year ago
  31. terenzreignz Group Title
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    That's right!

    • one year ago
  32. onegirl Group Title
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    ohh okay lol

    • one year ago
  33. onegirl Group Title
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    so thats the final answer?

    • one year ago
  34. terenzreignz Group Title
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    Well, you technically have two questions, but since there was an answer, then the integral converges, and it converges to 3 ^.^

    • one year ago
  35. onegirl Group Title
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    okay thanks

    • one year ago
  36. terenzreignz Group Title
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    No problem :)

    • one year ago
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