Evalaute: _0^2 4-x^2 d x - Vidyadip

Question

Evalaute:
$
\int_0^2 \sqrt{4-x^2} d x
$

✓

Answer

Let
$
\begin{aligned}
I & =\int_0^2 \sqrt{4-x^2} d x \\
& =\int_0^2 \sqrt{(2)^2-x^2} d x
\end{aligned}
$
We know that
$
\int \sqrt{a^2-x^2} d x=\frac{1}{2} x \sqrt{a^2-x^2}+\frac{1}{2} a^2 \sin ^{-1}\left(\frac{x}{a}\right)+C
$
So
$
I=\left(\frac{1}{2} x \sqrt{4-x^2}+\frac{1}{2} \times 4 \sin ^{-1}\left(\frac{x}{2}\right)\right)_0^2
$
$\begin{array}{c}=\left[0+2 \sin ^{-1}(1)-0\right]=2 \sin ^{-1} 1 \\ =2 \times \frac{\pi}{2}=\pi \text { }\end{array}$

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