Let be the area of the region (x,y) R^2: x^2+y^221, y^24x, x1.

Question

Let $\Delta$ be the area of the region $\{(x,y)\in\mathbb R^2: x^2+y^2\le21,\ y^2\le4x,\ x\ge1\}$. Then a certain combination of $\Delta$ and $21\sin^{-1}\left(\dfrac{2}{\sqrt7}\right)$ is equal to:

Accepted Answer

The parabola and circle meet at $x=3$ ($y=\pm2\sqrt3$). Splitting the region at $x=3$:

$$\Delta = \int_1^3 4\sqrt x\,dx + 2\int_3^{\sqrt{21}}\sqrt{21-x^2}\,dx = \frac83(3\sqrt3-1) + 21\sin^{-1}\!\left(\frac{2}{\sqrt7}\right) - 6\sqrt3.$$

*(The exact combination asked for in the stem was corrupted in the source; independent computation gives $\Delta-21\sin^{-1}(2/\sqrt7)=2\sqrt3-\dfrac83$, which does not exactly match any option --- the precise coefficient in the requested expression could not be recovered. The stated answer is option C.)*

Solutions · 1