Question

Find F · dr C for the given F and C. F = x2 i + y2 j and C is the top half of a circle of radius 3 starting at the point (3, 0) traversed counterclockwise. (a) Parameterize the path C in the usual way for circles. r (t) = (b) Set up the integral in terms of t only. F · dr C = dt (c) Evaluate the integral in part (b).

Answer 1

a. Take

$\vec r(t)=\sqrt3\cos t\,\vec\imath+\sqrt3\sin t\,\vec\jmath$

with $0\le t\le\pi$.

b. The line integral reduces to

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\int_0^\pi(3\cos^2t\,\vec\imath+3\sin^2t\,\vec\jmath)\cdot(-\sqrt3\sin t\,\vec\imath+\sqrt3\cos t\,\vec\jmath)\,\mathrm dt$

$=\displaystyle3\sqrt3\int_0^\pi(\sin^2t\cos t-\cos^2t\sin t)\,\mathrm dt$

c.

$\displaystyle3\sqrt3\left(\int_0^\pi\sin^2t\cos t\,\mathrm dt-\int_0^\pi\cos^2t\sin t\,\mathrm dt\right)$

$=\displaystyle3\sqrt3\left(\int_0^0u^2\,\mathrm du+\int_1^{-1}v^2\,\mathrm dv\right)$

(where $u=\sin t$ and $v=\cos t$)

$=\displaystyle-6\sqrt3\int_0^1v^2\,\mathrm dv=\boxed{-2\sqrt3}$