Question:

The area (in square units) of the region enclosed by the ellipse $x^2+3y^2=18$ in the first quadrant below the line $y=x/\sqrt{3}$ is:

$\sqrt{3}\pi-3$

$\sqrt{3}\pi+1$

$\sqrt{3}\pi$

$\sqrt{3}\pi+3$

Step-by-Step Solution

Key Concept: Line meets ellipse at $x=3$. Integrate $\int_0^3\frac{x}{\sqrt{3}}dx+\int_3^{3\sqrt{2}}\frac{\sqrt{18-x^2}}{\sqrt{3}}dx$.

Area $=\sqrt{3}\pi$.

Correct Answer: 3