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Question
Using integration find the area of the region:
\[\left\{ \left( x, y \right) : \left| x - 1 \right| \leq y \leq \sqrt{5 - x^2} \right\}\]
Solution
\[\left| x - 1 \right| \leq y \leq \sqrt{5 - x^2}\]
\[\left| x - 1 \right| = \sqrt{5 - x^2}\]
\[x = 2, - 1\]
\[A = \int_{- 1}^2 \left( \sqrt{5 - x^2} - \left| x - 1 \right| \right)dx\]
\[ = \int_{- 1}^2 \sqrt{5 - x^2} + \int_{- 1}^1 \left( x - 1 \right)dx + \int_1^2 \left( 1 - x \right)dx\]
\[ = \left[ \frac{x}{2}\sqrt{5 - x^2} + \frac{5}{2} \sin^{- 1} \left( \frac{x}{\sqrt{5}} \right) \right]_{- 1}^2 + \left[ \frac{x^2}{2} - x \right]_{- 1}^1 + \left[ x - \frac{x^2}{2} \right]_1^2 \]
\[ = \frac{5}{2}\left( \sin^{- 1} \left( \frac{2}{\sqrt{5}} \right) + \sin^{- 1} \left( \frac{1}{\sqrt{5}} \right) \right) + \frac{1}{2}\]
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