What parametrization should I use to evaluate $\int_{\phi}x^{4/3} + y^{4/3}$, where $\phi$ is curve given by $(x^2+y^2)^2 = 9(x^2-y^2)$?

Question

I´ve recently tried calculating this:

$$\int_{\phi}x^{4/3} + y^{4/3}$$ where $\phi$ is curve given by $(x^2+y^2)^2 = 9(x^2-y^2)$.

And I couldn´t think of any parametrization or substitution that would give me a reasonable outcome. Any suggestions?

Answer 1

The equation you are considering is the lemniscate of Bernoulli, and you can parametrise it as follows: \begin{align} x(t) &= 3 \frac{\cos(t)}{1+\sin^2(t)} \\ y(t) &= 3 \frac{\sin(t) \cdot \cos(t)}{1+\sin^2(t)} \end{align} for $t \in [0,2\pi]$.

As a further suggestion, note that to compute the line integral, you can use periodicity of the integrand to restrict the domain over which you need to integrate.