# Thread: line integral in a vector field

1. ## line integral in a vector field

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

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

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

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

2. you didn't differentiate r, it's dr not r dt

3. TRY this.......
Originally Posted by manyarrows
Given problem: Evaluate the line integral $\displaystyle \int_C \mathbf {F}\cdot d\mathbf r$, where C is given by the vector function $\displaystyle \mathbf r(t)$.

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

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

Thank the beagle