Question

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

Answer 1

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,
$$\begin{gathered} \mathrm{Required}\mathrm{area}={\int }_0^{\mathrm{\pi }}ydx \\ ={\int }_0^{\mathrm{\pi }}\left(\cos x\right)dx \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\cos x\right)dx+\left|{\int }_{\frac{\mathrm{\pi }}{2}}^{\mathrm{\pi }}\left(\cos x\right)dx\right| \\ ={\left(\sin x\right)}_0^{\frac{\mathrm{\pi }}{2}}+\left|{\left(\sin x\right)}_{\frac{\mathrm{\pi }}{2}}^{\mathrm{\pi }}\right| \\ =\left(\sin \frac{\mathrm{\pi }}{2}-\sin 0\right)+\left|\left(\sin \mathrm{\pi }-\sin \frac{\mathrm{\pi }}{2}\right)\right| \\ =\left(1-0\right)+\left|0-1\right| \\ =\left(1\right)+\left|-1\right| \\ =1+1 \\ =2 \\ \mathrm{Thus},\mathrm{Area}=2\mathrm{sq}.\mathrm{units} \end{gathered}$$


Hence, the area of the region bounded by the curve y = cosx between x = 0 and x = π is $\underline{2\mathrm{sq}.\mathrm{units}}.$