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theEric

Calculus 3 - Find work done along curve in force field. How do I set up the integral? The curve is the upper half of a circle with radius 3, centered at the origin, from (-3,0) to (3,0).

  • 11 months ago
  • 11 months ago

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  1. theEric
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    |dw:1367800601926:dw|

    • 11 months ago
  2. theEric
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    |dw:1367800740264:dw|

    • 11 months ago
  3. theEric
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    So I think it works nicely if I chose the parametric equations for the curve as\[x(t)=cos(t)\]\[y(t)=sin(t)\]and so\[\frac{dx(t)}{dt}=-sin(t)\]and\[\frac{dy(t)}{dt}=cos(t)\]

    • 11 months ago
  4. theEric
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    \[\text {So then the parameter, }t\text{, goes from }\pi \text{ to }0 \text{.}\] \[\text{Is it okay to write:} \int_\pi ^0 F(t)dt\text{, or must it be }-\int_0 ^ \pi F(t)dt \text{?}\] Don't they equal the same thing?

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  5. Zarkon
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    those two integrals are the same

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  6. Zarkon
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    you might want to look at your parametric equations again

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  7. theEric
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    \[\text{Thanks.. I've considered rewriting them to go from }0\text{ to }\pi \text{. In which case:}\] \[x(t)=-cos(t)\]\[y(t)=sin(t)\] Is this what you mean? Or were my original equations incorrect?

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  8. Zarkon
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    neither are correct

    • 11 months ago
  9. Zarkon
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    "circle with radius 3"

    • 11 months ago
  10. theEric
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    Oh... Yes.. Thank you! I tend to not carry all of the math into the next step. I tend to carry over my immediate thoughts and stop there...

    • 11 months ago
  11. theEric
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    \[x(t)=3cos(t)\]\[y(t)=3sin(t)\]\[\text{where }t \text{ goes from }0\text{ to }\pi \text{.}\] or\[x(t)=-3cos(t)\]\[y(t)=3sin(t)\]\[\text{where }t\text{ goes from }0 \text{ to } \pi \text{.}\]

    • 11 months ago
  12. theEric
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    The first one is from \[\pi\] to 0, I mean.

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  13. Zarkon
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    ok

    • 11 months ago
  14. theEric
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    Those work, then?

    • 11 months ago
  15. theEric
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    And so they're the same?

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  16. Zarkon
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    What is your force field \(F\)?

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  17. Zarkon
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    \[W=\int\limits_{a}^{b}F(c(t))\cdot c'(t) dt\]

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  18. Zarkon
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    \(c(t)\) is the particular path

    • 11 months ago
  19. theEric
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    To continue the problem I'm given,\[F(x,y)=(2y+x^2)i + (x^2 - 2x)j\]

    • 11 months ago
  20. theEric
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    yielding... \[W=-\int _0 ^\pi ((6sin(t)+9cos(t))(-3sint)+(9cos^2(t)-6cos(t))(3cos(t)))dt\]

    • 11 months ago
  21. theEric
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    Which is ugly.. I've simplified it a lot, and I think I've solved it, but I'm not sure. I'll just check with Wolfram Alpha. Thank you for clearing up the integration!

    • 11 months ago
  22. Zarkon
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    np

    • 11 months ago
  23. theEric
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    Take care!

    • 11 months ago
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