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

$x^2+y^2={\pi }^2$ represents a circle centered at $(0,0)$ with radius $\pi$.The area of the circle in the first quadrant (from $x=0$ to $x=\pi$) is:$A_1=\frac{1}{4}\pi r^2=\frac{1}{4}\pi (\pi {)}^2=\frac{{\pi }^3}{4}$ The area under the curve $y=\sin x$ from $x=0$ to $x=\pi$ is:$A_2={\int }_0^{\pi }\sin xdx=[-\cos x{]}_0^{\pi }=-(\cos \pi -\cos 0)=-(-1-1)=2$ The required area between the circle and the sine curve in the first quadrant is:Area $=A_1-A_2=\frac{{\pi }^3}{4}-2=\frac{{\pi }^3-8}{4}\text{ sq. units}$