Question

please help!!!!
Compute the line integral of the vector field F=<7y,-7x> over the circle x^2+y^2=36 oriented counterclockwise. Thanks!!!!

Answer 1

To compute \(\large \displaystyle \int_C \mathbf{F}\cdot d\mathbf{s} =\int_C 7y\,dx -7x\,dy\), you need to find a parameterization of \(\large C:x^2+y^2=36\) oriented counterclockwise. In particular, the parameterization you need to use is \(\large x=6\cos t\) and \(\large y=6\sin t\) for \(\large t\in[0,2\pi]\). Thus, \(\large \,dx = -6\sin t\,dt\) and \(\large \,dy= 6\cos t\,dt\).

Therefore,
\[\large \begin{aligned} \int_C 7y\,dx -7x\,dy &=\int_0^{2\pi}7(6\sin t)(-6\sin t\,dt )-7(6\cos t)(6\cos t\,dt)\\ &= \int_0^{2\pi}-252\sin^2t - 252 \cos^2 t\,dt\end{aligned}\]
I assume you can take things from here? :-)

Answer 2

Thank you so much! can I ask y another question

Answer 3

@ChristopherToni

Answer 4

Sure, ask away; (hopefully I can answer it! XD)