Question

Show the COMPLETE solution on the given problems.

1. Evaluate the line integral ∮2

Answer 1

Evaluate the line integral $\oint_C2ds$ where $C$ is the unit circle.

$$\oint_Cf(x, y)ds=\displaystyle\int_{a}^bf(x(t),y(t))\sqrt{\big({dx \over dt}\big)^2+\big({dy \over dt}\big)^2}dt$$

We first need parametric equations to represent $C.$ The unit circle can be parametrized by means of the equations

$$x=\cos t, y=\sin t, \ 0\leq t\leq2\pi.$$

$${dx \over dt}=-\sin t,\ {dy \over dt}=\cos t$$

$$\sqrt{\big({dx \over dt}\big)^2+\big({dy \over dt}\big)^2}=\sqrt{\big(-\sin t\big)^2+\big(\cos t\big)^2}=1$$

$$\oint_C2ds=\displaystyle\int_{0}^{2\pi}2(1)dt=2\big[t\big]\begin{matrix} 2\pi \\ 0 \end{matrix}=2(2\pi-0)=4\pi$$