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anonymous

  • one year ago

What is wrong with the following calculation? int_{-2}^{1}\frac{ 1 }{ x^4 }dx=\frac{ -1 }{ 3x^3 }]^1(-2)below=\frac{ -1 }{ 3 }-\frac{ 1 }{ 24 }=\frac{ -9 }{ 24 }

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  1. anonymous
    • one year ago
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    This is the question. \[\int\limits_{-2}^{1}\frac{ 1 }{ x^4 }dx\]

  2. Zarkon
    • one year ago
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    what happens when x=0

  3. anonymous
    • one year ago
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    it would be undefined?

  4. Zarkon
    • one year ago
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    so this integral is an improper integral

  5. anonymous
    • one year ago
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    what? I don't quite understand...

  6. Zarkon
    • one year ago
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    was all that work above given and you are to tell why it is wrong?

  7. Zarkon
    • one year ago
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    or is it your work?

  8. anonymous
    • one year ago
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    It's not my work, it's from the textbook

  9. Zarkon
    • one year ago
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    to use the FTC the integrand (the thing inside the integral) must be continuous on the interval you are integrating over

  10. anonymous
    • one year ago
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    okay...

  11. zepdrix
    • one year ago
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    still confused? :o

  12. anonymous
    • one year ago
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    yes... :(

  13. zepdrix
    • one year ago
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    1/x^4 has an asymptote at x=0, ya? :o

  14. anonymous
    • one year ago
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    yes

  15. zepdrix
    • one year ago
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    so our function is not continuous over the given interval.|dw:1433734647150:dw|See the problem the 0 is creating for us? +_+

  16. anonymous
    • one year ago
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    Oooh I see... I think I got now.. Thanks!

  17. zepdrix
    • one year ago
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    If you want to be more "mathy" about it, you can break up the integral at that 0 point,\[\Large\rm \int\limits_{-2}^1 \frac{1}{x^4}dx=\int\limits_{-2}^0 \frac{1}{x^4}dx+\int\limits_{0}^1 \frac{1}{x^4}dx\]But it's still improper since our function isn't defined at zero, so we introduce some limits,\[\Large\rm =\lim_{b\to 0}\int\limits\limits_{-2}^b \frac{1}{x^4}dx+\lim_{b\to0}\int\limits\limits_{b}^1 \frac{1}{x^4}dx\]And then uhhh, in your calculuations you should end up with some stuff... \(\Large\rm \infty\), that kinda stuff. Which will show you that it doesn't converge. I dunno, I like thinking about the graph better :P

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