I need to find the area of the middle part bounded (or between) 2 curves:
$ x²+y²=1$ and $ 4x²-y²=1$.
I have the graphic of the middle part (the part, which I need to calculate the area for it), but I can't understand, do I need to solve this in polar system or Cartesian?
I think, that I can move to Cartesian, but I don't know which function to integrate.
In Cartesian system I'm solving this integral:
$$\int_{-\frac{\sqrt{1+y^2}}{2}}^{\frac{\sqrt{1+y^2}}{2}}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}1\ dx\ dy$$
But I'm getting a function of x or y, not a number.
And in polar system I can't understand which function to use, I know that $\ r$ will vary from $\ 0$ to $\ 1$ and $\theta$ will vary from from $0$ to $\ 2\ π$.
Best Answer
Hint: The area of one of the unshaded parts is given by
$$\int_{-\sqrt{\tfrac35}}^{\sqrt{\tfrac35}} \int_{\tfrac{\sqrt{1+y^2}}2}^{\sqrt{1-y^2}} dx \, dy$$
which can be used to determine the desired area.