Find appropriate parametrization to find area bounded by a curve

[wifi]

Problem:

Use an appropraite parametrization $$x=f(r,\theta), y=g(r,\theta)$$ and the corresponding Jacobian such that $$dx \ dy \ =|J| dr \ d\theta$$ to find the area bounded by the curve $$x^{2/5}+y^{2/5}=a^{2/5}$$

Attempt at a Solution:

I'm not really sure how to find the parametrization. Once I have that, calculating the Jacobian is simple. Then all that's left is computing a double integral. Right?

 

[STEMucator]

Problem:

Use an appropraite parametrization $$x=f(r,\theta), y=g(r,\theta)$$ and the corresponding Jacobian such that $$dx \ dy \ =|J| dr \ d\theta$$ to find the area bounded by the curve $$x^{2/5}+y^{2/5}=a^{2/5}$$

Attempt at a Solution:

I'm not really sure how to find the parametrization. Once I have that, calculating the Jacobian is simple. Then all that's left is computing a double integral. Right?

Hint: Your parametrization hints at polar co-ordinates.

$x = rcos\theta, y = rsin\theta$

 

[wifi]

But if I use $x=rcos\theta$ and $y=rsin\theta$, then I have $dx=rd\theta$ and $dy=rd\theta$. Also $|J|=r$. However we want $dx \ dy=|J|dr \ d\theta$.

Using $x=rcos\theta$ and $y=rsin\theta$, then the curve is given by $$(rcos\theta)^{\frac{2}{5}}+(rsin\theta)^{\frac{2}{5}}=a^{\frac{2}{5}}$$

So to find the area bounded by this curve we want to perform a double integral. I'm just confused on setting it up. Now that we're in terms of $r$ and $\theta$, how do we figure out the limits of integration for the given curve?