A woman discovered a scratch along a straight line on a circular table top of radius 8 cm. She divided the table top into 4 equal quadrants and…

Equation of the circular tabletop: $\displaystyle x^2 + y^2 = 64$Equation of line (scratch): $\displaystyle x = y$The line and circle intersect at $\displaystyle x = 4\sqrt{2}$Area of the shaded region$$= \int_{0}^{4\sqrt{2}} x\,dx + \int_{4\sqrt{2}}^{8} \sqrt{64 - x^2}\,dx$$$$= \left.\frac{x^2}{2}\right]_{0}^{4\sqrt{2}} + \left[\frac{x}{2}\sqrt{64 - x^2} + 32\sin^{-1}\frac{x}{8}\right]_{4\sqrt{2}}^{8}$$$$= \frac{32}{2} + 32\sin^{-1}1 - 2\sqrt{2}\cdot 4\sqrt{2} - 32\sin^{-1}\frac{1}{\sqrt{2}}$$$$= 16 + 16\pi - 16 - 8\pi = 8\pi\ \text{cm}^2$$