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Question

Area of the region bounded by the curve y = cos x between x = 0 and x = π is _______________.

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Solution

To find: area of the region bounded by the curve y = cosx between x = 0 and x = π The required area of the region = ${\int }_{0}^{\mathrm{\pi }}ydx$ Thus, $\mathrm{Required}\mathrm{area}={\int }_{0}^{\mathrm{\pi }}ydx\phantom{\rule{0ex}{0ex}}={\int }_{0}^{\mathrm{\pi }}\left(\mathrm{cos}x\right)dx\phantom{\rule{0ex}{0ex}}={\int }_{0}^{\frac{\mathrm{\pi }}{2}}\left(\mathrm{cos}x\right)dx+\left|{\int }_{\frac{\mathrm{\pi }}{2}}^{\mathrm{\pi }}\left(\mathrm{cos}x\right)dx\right|\phantom{\rule{0ex}{0ex}}={\left(\mathrm{sin}x\right)}_{0}^{\frac{\mathrm{\pi }}{2}}+\left|{\left(\mathrm{sin}x\right)}_{\frac{\mathrm{\pi }}{2}}^{\mathrm{\pi }}\right|\phantom{\rule{0ex}{0ex}}=\left(\mathrm{sin}\frac{\mathrm{\pi }}{2}-\mathrm{sin}0\right)+\left|\left(\mathrm{sin}\mathrm{\pi }-\mathrm{sin}\frac{\mathrm{\pi }}{2}\right)\right|\phantom{\rule{0ex}{0ex}}=\left(1-0\right)+\left|0-1\right|\phantom{\rule{0ex}{0ex}}=\left(1\right)+\left|-1\right|\phantom{\rule{0ex}{0ex}}=1+1\phantom{\rule{0ex}{0ex}}=2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Thus},\mathrm{Area}=2\mathrm{sq}.\mathrm{units}$ Hence, the area of the region bounded by the curve y = cosx between x = 0 and x = π is $\overline{)2\mathrm{sq}.\mathrm{units}}.$

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