Question

Which antiderivative is used in evaluating \[\int \sqrt{a^2-x^2}\,dx\] for the ellipse area?

Options

• \[\frac{x^2}{2}\sqrt{a^2-x^2}\]

• \[\frac{a}{2}\sin^{-1}\frac{x}{b}\]

• \[\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}\]

• \[\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}\]

Answer 1

The required integration formula is \[\int\sqrt{a^2-x^2}\,dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}\]. It is evaluated at the limits \[0\] and \[a\].