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

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

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

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

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