Question

What is the area enclosed by the ellipse \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\]?

Options

• \[\pi(a+b)\]

• \[\pi ab\]

• \[\frac{\pi ab}{2}\]

• \[2\pi ab\]

Answer 1

Evaluating \[4\int_0^a \frac{b}{a}\sqrt{a^2-x^2}\,dx\] gives \[\frac{4b}{a}\cdot\frac{a^2}{2}\cdot\frac{\pi}{2}\]. This simplifies to \[\pi ab\].