So I'm asked to evaluate the line integral
. . . . .\(\displaystyle \displaystyle \int_C\, F\, dr \)
where
. . . . .\(\displaystyle F\, =\, yz\vec{i}\, +\, siny\vec{j}\, +\, cosz \vec{k} \)
along the curve
. . . . .\(\displaystyle r(t)\,=\,t^2\vec{i}\,+\,t\vec{j}\,+\,t^3\vec{k} \)
for \(\displaystyle 1\, \leq \,t\, \leq\, 3 \)
My guess would be to find some kind of parametization for \(\displaystyle F\) in terms of t, and then apply the forumla
. . . . .\(\displaystyle \displaystyle \int_C\, F\, dr\, =\, \int_C\, F(r(t))\, \cdot\, r'(t)\,dt\)
But I can't seem to find such a parametization. Once I do find one, I don't think it'd be too hard to just plug \(\displaystyle F(r(t))\) and \(\displaystyle r'(t)\) into the forumla.
. . . . .\(\displaystyle \displaystyle \int_C\, F\, dr \)
where
. . . . .\(\displaystyle F\, =\, yz\vec{i}\, +\, siny\vec{j}\, +\, cosz \vec{k} \)
along the curve
. . . . .\(\displaystyle r(t)\,=\,t^2\vec{i}\,+\,t\vec{j}\,+\,t^3\vec{k} \)
for \(\displaystyle 1\, \leq \,t\, \leq\, 3 \)
My guess would be to find some kind of parametization for \(\displaystyle F\) in terms of t, and then apply the forumla
. . . . .\(\displaystyle \displaystyle \int_C\, F\, dr\, =\, \int_C\, F(r(t))\, \cdot\, r'(t)\,dt\)
But I can't seem to find such a parametization. Once I do find one, I don't think it'd be too hard to just plug \(\displaystyle F(r(t))\) and \(\displaystyle r'(t)\) into the forumla.
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