There is a hollow hemisphere $x^2+y^2+z^2=4$ $(z<0)$ in the coordinate space, and in the $xy$-plane, there is a square whose center is the origin and whose length is $2$ on one side. When you illuminate the square with a point light source at point $(0,0,2)$, it will form a shadow on the hemisphere's surface. The question is to calculate the area of the shadow.
My teacher's hint was: We can approximately calculate a curved surface's area if the width is small enough, it can be regardes as the product of the length of the band and the length of the width. And calculate $$\int_1^{\sqrt2}{\frac{1}{(r^2+4)r(\sqrt{r^2-1})}\,\mathrm{d}r}=0.15.$$
I tried to set the square on the $xy$-plane and wlog $(1,1,0),(1,-1,0),(-1,1,0),(-1,-1,0)$, but I can't even imagine how to get started.
