Cambridge IGCSE Additional Mathematics Challenge Q This is a challenge question from my Cambridge IGCSE Additional Maths textbook. Bear with me on the drawing. The drawing consists of a square, circle, and quarter circle. The only measurement give is that the side length of the square is $10$cm. Can someone help me find the area of the shaded region? I am looking for an explanation of the answer as well.


Hint. Let the $x$-axis run along the diagonal from bottom left to top right. Then the equation of the small circle is $x^2+y^2=5^2$ and that of the big circle $x^2+(y+\sqrt{50})^2=10^2.$ The two intersect at the points $$\left(\pm\frac{5\sqrt 7}{2\sqrt2},\frac{5}{2\sqrt2}\right).$$
Thus the area is given by $$2\int_0^{5\sqrt 7/2\sqrt 2}\left(\sqrt{5^2-x^2}-\sqrt{10^2-x^2}+5\sqrt{2}\right)\mathrm dx.$$
Can you now proceed?

Based on the fact that OP may not know calculus, as hinted at in the comments, I add that the integral evaluates to $$25\left(\alpha-4\beta+\frac{\sqrt 7}{2}\right),$$ where $\cos\alpha=1/2\sqrt 2,\,\cos\beta=5/4\sqrt 2,$ and the acute angles $\alpha,\,\beta$ are in radians.
