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Math Help - Line Integral

  1. #1
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    Line Integral

    c(t) = (cos^3(t),sin^3(t)) t goes from 0 to 2pi
    F(x,y) = xi + yj

    I know the equations but something tells me if i do the gradient it equals 0. The problem I'm having is how to do and find the gradient.
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  2. #2
    MHF Contributor matheagle's Avatar
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    you still need to say that you want integrated

    maybe \int_C  F\cdot dC
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  3. #3
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    Why gradient? That is the opposite of the integral. I presume that you mean you are to integrate \int_C F(x,y)\cdot d\vec{s}= \int_C xdx+ ydy. If C(t)= cos^3(t)\vec{i}+ sin^3(t)\vec{j} then d\vec{s}= -3sin(t)cos^2(t)dt\vec{i}+ 3sin^2(t)cos(t)dt\vec{j} or dx= -3sin(t)cos^2(t)dt and dy= 3sin^2(t)cos(t)dt

    F(x,y)= cos^3(t)\vec{i}+ sin^3(t)\vec{j} so you integral is
    \int_0^{2\pi} (-3sin(t)cos^5(t)+ 3sin^5(t)cos(t))dt

    Since that involves only odd powers of sine and cosine, it should be easy to integrate.
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