Question

​The area of the bounded by the curve y = sin x between the ordinates x = 0, $x=\frac{\mathrm{\pi }}{2}$ and the x-axis is _____________.

Answer 1

To find: area of the region bounded by the curve y = sin x between the ordinates x = 0, $x=\frac{\mathrm{\pi }}{2}$ and the x-axis

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


Hence, the area of the region bounded by the curve y = sin x between the ordinates x = 0, $x=\frac{\mathrm{\pi }}{2}$ and the x-axis is $\underline{1\mathrm{sq}.\mathrm{unit}}.$