1. ## Line integral

Let $x = \cos^{3} t$ and $y = \sin^{3}t$ ( $0 \leq t \leq 2 \pi$). Also $\rho(x,y) = k$.

Find $I_0 = \int_{C} (x^{2} + y^{2}) \ dm$

So $m = \int_{C} k \ ds = 3k \int_{0}^{2 \pi} \cos t \sin t \ dt$.

Then $dm = 3k \cos t \sin t \ dt$.

So does $I_0 = 3k\int_{0}^{2 \pi} \left( \cos^{6} t + \sin^{6} t \right)(\cos t \sin t) \ dt = 0$

2. So if the work is 0 does this imply that the path is closed?

3. I don't think it physically makes sense for $I_0 = 0$?