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Thread: How to simplify this (line) integral?

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

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

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


    I find that I have to solve the integral:

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

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

    $\displaystyle \dfrac{\partial }{\partial y}(y)=\dfrac{\partial }{\partial x}(x)$
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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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    $\displaystyle \vec{F}$ is a conservative field i.e. there exists a scalar field $\displaystyle f$ such that $\displaystyle \nabla f=\vec{F}$.

    By a well known theorem,

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

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