Question

\[\int\limits_0^2 \sqrt{4 - x^2} dx\]

Answer 1

\[\int_0^2 \sqrt{4 - x^2} d x\]
\[ = \int_0^2 \sqrt{2^2 - x^2} d x\]
\[ = \left[ \frac{x}{2}\sqrt{4 - x^2} + \frac{1}{2} \times 2^2 \sin^{- 1} \frac{x}{2} \right]_0^2 \]
\[ = \left[ \frac{x}{2}\sqrt{4 - x^2} \right]_0^2 + 2 \left[ \sin^{- 1} \frac{x}{2} \right]_0^2 \]
\[ = 0 + 2\left( \frac{\pi}{2} - 0 \right)\]
\[ = \pi\]