Let A₁ be the area bounded by the curves x² + y² = 25, 4y = | 4 – x²| and x = 0 lying in the first quadrant and A₂ be the remaining region in the first quadrant then

$\text{ The equation }{x}^2+y^2=25\text{ represents a circle of radius }5\text{ with centre at }(0,0)$

$\text{ the two curve intersect each other at x=±4}$

$\text{ required area }={\int }_0^4 \left(\sqrt{25-x^2}-\left|1-\frac{x^2}{4}\right|\right)dx$

${\mathrm{A}}_1={\int }_0^4 \sqrt{25-{\mathrm{x}}^2}\mathrm{dx}-{\int }_0^2 \frac{1}{4}\left(4-{\mathrm{x}}^2\right)\mathrm{dx}-{\int }_2^4 \frac{1}{4}\left({\mathrm{x}}^2-4\right)\mathrm{dx}$

$A_1=2+\frac{25}{2}{\tan }^{-1}\frac{4}{3}$

$A_1=2+\frac{25}{2}\left(\frac{\mathrm{\pi }}{2}-{\tan }^{-1}\frac{3}{4}\right)=2+\frac{25\mathrm{\pi }}{4}-\frac{25}{2}{\tan }^{-1}\frac{3}{4}$