# Let F(x,y)=(:-y,x:) and C be the ellipse (x^2)/(16)+(y^2)/(9)=1 oriented counter clockwise, then find the value of int_C F.dr

Averi Fields 2022-09-25 Answered
Let $F\left(x,y\right)=⟨-y,x⟩$ and C be the ellipse $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{9}=1$ oriented counter clockwise, then find the value of ${\int }_{C}F.dr$
This is how I tried it,
$x=4\mathrm{cos}\left(t\right)\to dx=-4\mathrm{sin}\left(t\right)$
$y=3\mathrm{sin}\left(t\right)\to dy=3\mathrm{cos}\left(t\right)$
and $0\le t\le 2\pi$
${\int }_{C}F.dr=\int -ydx+xdy={\int }_{t=0}^{2\pi }12\left({\mathrm{sin}}^{2}\left(t\right)+{\mathrm{cos}}^{2}\left(t\right)\right)dt=75.4$
But answer given to me at the back is 98.2 and doesn't agrees with mine.
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## Answers (1)

skarvama
Answered 2022-09-26 Author has 5 answers
Your solution is correct. In fact,
${\int }_{C}F\left(r\right)\cdot dr={\int }_{0}^{2\pi }\left(-3\mathrm{sin}t,4\mathrm{cos}t\right)\cdot \left(-4\mathrm{sin}t,3\mathrm{cos}t\right)dt={\int }_{0}^{2\pi }12\left({\mathrm{sin}}^{2}t+{\mathrm{cos}}^{2}t\right)dt=24\pi \approx 75.3982$
There is a typo/mistake in the book. It happens sometimes.
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