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Explain solution RD Sharma class 12 chapter 19 Definite Integrals exercise Very short answer type question 24 maths

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Answer: \pi

Hint: you must know the rule of integration

Given:\int_{0}^{2} \sqrt{4-x^{2}}

Solution:  \int_{0}^{2} \sqrt{4-x^{2}}

\begin{aligned} &\text { Put } x=2 \sin \theta \\\\ &d x=2 \cos \theta d \theta \end{aligned}

       \begin{aligned} &=\int_{0}^{\frac{\pi}{2}} \sqrt{4-4 \sin ^{2} \theta} \cdot 2 \cos \theta d \theta \\\\ &=\int_{0}^{\frac{\pi}{2}} 4 \cos ^{2} \theta d \theta \end{aligned}

       \begin{aligned} &=\int_{0}^{\frac{\pi}{2}} 4 \cdot \frac{1}{2}(1+\cos 2 \theta) d \theta \\\\ &=2\left[\theta+\frac{\sin 2 \theta}{2}\right]_{0}^{\frac{\pi}{2}} \end{aligned}

       \begin{aligned} &=2\left[\frac{\pi}{2}+\frac{\sin \pi}{2}-0-0\right] \\\\ &=\pi \end{aligned}




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