Question

The area of the bounded by the curve y = sinx, x-axis and between x = 0 and x = 2π is _______________.

Answer 1

To find: area of the region bounded by the curve y = sinx, x-axis and between x = 0 and x = 2π



The required area of the region = ${\int }_0^{2\mathrm{\pi }}ydx$
Thus,
$$\begin{gathered} \mathrm{Required}\mathrm{area}={\int }_0^{2\mathrm{\pi }}ydx \\ ={\int }_0^{2\mathrm{\pi }}\left(\sin x\right)dx \\ ={\int }_0^{\mathrm{\pi }}\left(\sin x\right)dx+\left|{\int }_{\mathrm{\pi }}^{2\mathrm{\pi }}\left(\sin x\right)dx\right| \\ ={\left(-\cos x\right)}_0^{\mathrm{\pi }}+\left|{\left(-\cos x\right)}_{\mathrm{\pi }}^{2\mathrm{\pi }}\right| \\ =\left(-\mathrm{cos\pi }-\left(-\cos 0\right)\right)+\left|\left(-\cos 2\mathrm{\pi }-\left(-\mathrm{cos\pi }\right)\right)\right| \\ =\left(1+\cos 0\right)+\left|-1+\mathrm{cos\pi }\right| \\ =\left(1+1\right)+\left|-1-1\right| \\ =2+\left|-2\right| \\ =2+2 \\ =4 \\ \mathrm{Thus},\mathrm{Area}=4\mathrm{sq}.\mathrm{units} \end{gathered}$$


Hence, the area of the region bounded by the curve y = sinx, x-axis and between x = 0 and x = 2π is $\underline{4\mathrm{sq}.\mathrm{units}}.$