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nathanruff

  • 2 years ago

Use the fact that the function is odd to show that the inequality is true. (Anyone who helps me will receive a medal because I'm stuck.)

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  1. nathanruff
    • 2 years ago
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    I already verified that the function is odd. I just need help showing that: \[0 \le \int\limits_{-2}^{3}\sin \sqrt[3]{x}dx \le 1\]

  2. nathanruff
    • 2 years ago
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    Using the fact that the function is odd I know that\[0 \le \int\limits_{-2}^{3}\sin \sqrt[3]{x}dx \le 1 = 0 \le \int\limits_{2}^{3}\sin \sqrt[3]{x}dx \le 1\]

  3. nathanruff
    • 2 years ago
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    Now, I just need to show that \[0 \le \int\limits_{2}^{3}\sin \sqrt[3]{x}dx \le 1\]

  4. mark_o.
    • 2 years ago
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    is it like this \[\sin ^{2}(\sqrt{x}) ?\]

  5. nathanruff
    • 2 years ago
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    It's actually\[\sin (\sqrt[3]{x})\] or \[\sin(x^{1/3})\]

  6. mark_o.
    • 2 years ago
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    what did you get on this one? \[\int\limits_{}^{}\sin \sqrt{x}dx= --------------?\]

  7. nathanruff
    • 2 years ago
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    \[= \cos \sqrt{x} + C\]Is that correct?

  8. mark_o.
    • 2 years ago
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    no, you need to consider (x)^1/2=u,.....

  9. nathanruff
    • 2 years ago
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    Ah, u substitution.

  10. nathanruff
    • 2 years ago
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    \[= -\cos \sqrt{x}+C\] Wouldn't that be it?

  11. mark_o.
    • 2 years ago
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    if we let u=x^1/2 du=1/(2x^1/2)dx

  12. mark_o.
    • 2 years ago
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    youll come up to = 2 sin x^1/2 -2 x^1/2 cos x^1/2 ] from 2 to 3

  13. mark_o.
    • 2 years ago
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    is it from -2 to 3?

  14. nathanruff
    • 2 years ago
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    Oh, okay. To begin with, it's from -2 to 3, but the integral from -2 to 0 and the integral from 0 to 2 cancel each other out this the function is odd and symmetric about the origin, so in its simplest form, the integral is from 2 to 3.

  15. mark_o.
    • 2 years ago
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    do you have a calculator that does derivative and integrals? that way you can re check your work... :D

  16. nathanruff
    • 2 years ago
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    Yep! I got my handy-dandy TI-84 Plus. ;)

  17. mark_o.
    • 2 years ago
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    ok good. did you have the same answer?

  18. mark_o.
    • 2 years ago
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    ok good. did you have the same answer from your calculator from 2 to 3 ??

  19. nathanruff
    • 2 years ago
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    Yep. Thank you!

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