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Math Help - line integral in a vector field

  1. #1
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    line integral in a vector field

    Given problem: Evaluate the line integral \int_C \mathbf {F}\cdot d\mathbf r, where C is given by the vector function \mathbf r(t).

    \mathbf F(x,y,z)=\sin x\mathbf i+\cos y\mathbf j+xz\mathbf k
    \mathbf r(t)=t^3\mathbf i-t^2\mathbf j+t\mathbf k
    0\leq t\leq 1

    My work:
    x=t^3, y=t^2, z=t
    \int^1_0 [\sin (t^3)\mathbf i+\cos(-t^2)\mathbf j+(t^3)(t)\mathbf k]\cdot[t^3\mathbf i-t^2\mathbf j+t\mathbf k]dt
    \int^1_0 t^3\sin(t^3)-t^2\cos(-t^2)+t^5 dt

    I have tried integration by parts and substitution and none work on first two terms. I am guessing I have messed up in the set up above since it should be a non intensive problem (i'm only in calc 3)

    Thanks for any help
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  2. #2
    MHF Contributor matheagle's Avatar
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    you didn't differentiate r, it's dr not r dt
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  3. #3
    MHF Contributor matheagle's Avatar
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    TRY this.......
    Quote Originally Posted by manyarrows View Post
    Given problem: Evaluate the line integral \int_C \mathbf {F}\cdot d\mathbf r, where C is given by the vector function \mathbf r(t).

    \mathbf F(x,y,z)=\sin x\mathbf i+\cos y\mathbf j+xz\mathbf k
    \mathbf r(t)=t^3\mathbf i-t^2\mathbf j+t\mathbf k
    0\leq t\leq 1

    Beagle work:
    x=t^3, y=t^2, z=t
    \int^1_0 [\sin (t^3)\mathbf i+\cos(-t^2)\mathbf j+(t^3)(t)\mathbf k]\cdot[3t^2\mathbf i-2t\mathbf j+\mathbf k]dt
    \int^1_0 \left[3t^2\sin(t^3)-2t\cos(-t^2)+t^4\right] dt


    Thank the beagle
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