Find the area of $S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$ Let $S$ be the domain defined by $$S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$$
find the area of $S$
This is middle school problem,so I think it can be solved without integral methods?
$$(x-\sin^2{\theta})^2+(y-\cos^2{\theta})^2=1$$

From $(x - \sin^2\theta)^2 + (y - \cos^2\theta)^2 = 1$ we get that $S$ is a set of circles with radius $1$ and centers on the points of $(\sin^2\theta, \cos^2\theta)$, i.e. on the segmet from point $(1,0)$ to point $(0,1)$ (see figure below).
So the area of $S$ is sum of areas of two semicircles and area of rectngle with sides lengths $2$ and $\sqrt{2}$: 
$$
\frac{1}{2}\pi\cdot 1^2 + 2\cdot\sqrt{2} + \frac{1}{2}\pi\cdot 1^2 = \pi + 2\sqrt{2}.
$$

