Question

The area in the first quadrant between $x^2+y^2={\mathrm{\pi }}^2$ and$y=\sin x$ is

• A. $\frac{{\mathrm{\pi }}^3-8}{4}$
• B. $\frac{{\mathrm{\pi }}^3}{4}$
• C. $\frac{{\mathrm{\pi }}^3-16}{4}$
• D. $\frac{{\mathrm{\pi }}^3-8}{2}$

Answer 1

The correct option is A

$\frac{{\mathrm{\pi }}^3-8}{4}$

Explanation for correct answer:

Determining the area in the first quadrant between the given functions.

Given equation of curves are shown as follows:

$x^2+y^2={\mathrm{\pi }}^2\to \left(\mathrm{i}\right)$

$y=\sin x\to \left(ii\right)$

The first equation represents the circle of radius '$\mathrm{\pi }$' units.

The area of a shaded region is given by.

Required area=Area of circle in first quadrant - area of first half of Sine curve

$$\begin{gathered} A=\frac{{\mathrm{\pi r}}^2}{4}+{\int }_0^{\mathrm{\pi }}\sin \left(x\right)dx \\ =\frac{\pi \times {\mathrm{\pi }}^2}{4}+{\left[\cos \left(x\right)\right]}_0^{\mathrm{\pi }}\left[\because r=\mathrm{\pi }\right] \\ =\frac{{\mathrm{\pi }}^3}{4}+\left[\cos \left(\mathrm{\pi }\right)-\cos \left(0\right)\right] \\ =\frac{{\mathrm{\pi }}^3}{4}+\left[-1-1\right] \\ =\frac{{\mathrm{\pi }}^3}{4}-2 \\ =\frac{{\mathrm{\pi }}^3-8}{4} \end{gathered}$$

Therefore the area of the shaded portion is equal to $\frac{{\mathrm{\pi }}^3-8}{4}squareunits$

Hence, the correct answer option (A).