Evaluate the following integrals: ^2_0 4-x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^2_0\sqrt{4-\text{x}^2}\text{ dx}$

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Answer

$\int\limits^2_0\sqrt{4-\text{x}^2}\text{ dx}$
$=\int\limits^2_0\sqrt{2^2-\text{x}^2}\text{ dx}$
$=\Big[\frac{\text{x}}{2}\sqrt{4-\text{x}^2}\text{ dx}+\frac{1}{2}\times2^2\sin^{-1}\frac{\text{x}}{2}\Big]^2_0$
$=\Big[\frac{\text{x}}{2}\sqrt{4-\text{x}^2}\text{ dx}\Big]^2_0+2\Big[\sin^{-1}\frac{\text{x}}{2}\Big]^2_0$
$=0+2\Big(\frac{\pi}{2}-0\Big)$
$=\pi$

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