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anonymous

  • one year ago

A bee with velocity vector r'(t) starts out at the origin at t=0 and flies around for T seconds. Where is the bee located at time T if ∫(0-T) r'(u)du = 0? Also, what does the quantity ∫(0-T) ||r'(u)||du represent?

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  1. anonymous
    • one year ago
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    Please help

  2. amistre64
    • one year ago
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    what does (0-T) represent?

  3. anonymous
    • one year ago
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    integral from 0 to T

  4. anonymous
    • one year ago
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    sorry it's the only way I know how to type it

  5. amistre64
    • one year ago
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    so those are the limits, ok

  6. amistre64
    • one year ago
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    do you recall the definition of: \[\int_{a}^{b}f(x)~dx=??\]

  7. amistre64
    • one year ago
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    the fundamental thrm of calculus is also what it is called

  8. anonymous
    • one year ago
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    It's F(b) - F(a)

  9. amistre64
    • one year ago
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    correct, so if F(b)-F(a) = 0, then either a=b, or F has the same value at multiple input levels similar to a periodic function.

  10. amistre64
    • one year ago
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    for example, cos(pi) - cos(0) = 0 since cos(pi)=cos(0)

  11. amistre64
    • one year ago
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    but the question is, what does taking the integral of a function tell us?

  12. anonymous
    • one year ago
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    it can tell you what the area under a curve is

  13. amistre64
    • one year ago
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    and my example is spose to read cos(2pi) = cos(0), but the site is lagging for me

  14. anonymous
    • one year ago
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    it's ok

  15. amistre64
    • one year ago
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    the area under a curve, yes ... but that reflects displacement

  16. amistre64
    • one year ago
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    ever wonder why in some cases we get a negative area? and in other cases we dont?

  17. anonymous
    • one year ago
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    Yes because the integral of velocity is displacement

  18. anonymous
    • one year ago
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    right?

  19. anonymous
    • one year ago
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    negative displacement ?

  20. amistre64
    • one year ago
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    when we want to find the area between 2 curves, area is a positive value always ... its an absolute value. And in those cases we have to concern ourselves with the places that one curve crosses over the other.

  21. amistre64
    • one year ago
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    displacement on the other hand reflects is a vector, it has magnitude and direction.

  22. amistre64
    • one year ago
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    F(b)-F(a) = 0 assumes that at a and b, we are in the same place. Either a = b, or we have simply ended were we started.

  23. anonymous
    • one year ago
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    Then the location of the bee is still the origin at time T?

  24. amistre64
    • one year ago
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    yes, at time T, the bee is simply back in the same place it was at when the time was equal to 0

  25. anonymous
    • one year ago
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    But why substitute u in for t for the integration? That's the most confusing part. Does that have any effect on the result?

  26. amistre64
    • one year ago
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    0 and T are values that the variable u can take on. the function itself is dependant on the value of u ...

  27. anonymous
    • one year ago
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    What is the difference between ∫(0-T) r'(u)du = 0 and ∫(0-T) r'(t)dt = 0 Why did they substitute ?

  28. amistre64
    • one year ago
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    hmm, im not absolutely sure, but i think it has something to do with what is called a 'dummy' variable - meant to avoid confusion between the overuse of the letter T

  29. anonymous
    • one year ago
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    oh ok .. well thank you !

  30. amistre64
    • one year ago
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    good luck

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