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Math Help - How to simplify this (line) integral?

  1. #1
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    How to simplify this (line) integral?

    The curve is C parametrized by  \vec{r}(t) = (t^4cos^2(2\pi e^t),t^4sin^2(2\pi e^t)) , where 0 \leq t \leq \pi /4. We also have  \vec{F} (x,y) = (y,x)

    Calculate the line integral: \int_C \vec{F}* d\vec{r}


    I find that I have to solve the integral:

    \int_0^{\pi/4} {(8t^7cos(t)sin(t) + t^8(cos^2(t)-sin^2(t))) dt}

    How can I solve this integral quickly?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint

    \dfrac{\partial }{\partial y}(y)=\dfrac{\partial }{\partial x}(x)
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  3. #3
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    I don't understand the hint
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    \vec{F} is a conservative field i.e. there exists a scalar field f such that \nabla f=\vec{F}.

    By a well known theorem,

    \displaystyle\int_C\vec{F}\cdot\vec {r}=f(B)-f(A)

    being A,B respectively the initial and final points of C. So, you only need to find f.
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