What is the area of a shape, which I'm deeming a 'cylicoid', which is defined as follows: Circle A of radius 1 is held stationary. Circle B of radius 1 has a point on its rim which traces a path as it is rolled around circle A. A cyclicoid is the shape enclosed by the path of the point on circle B's rim. It looks roughly like a heart shape.
In parametric terms, a cyclicoid can be shown to be:
x = 2 cos t + cos (2t)
y = 2 sin t + sin (2t)
However, I'm unsure how to integrate this parametric function to find its area. Any ideas on how to do it and what the answer is? Thanks.
By stokes' theorem, if $\gamma$ is a closed curve in the plane, then the area it encloses is given by$$S=\int dxdy=\int_\gamma ydx.$$