1.31 Integrals associated with conics

For positive constants $a$ and $b$, consider the curves:

(i) parabola $y^{2}=ax+b$;

(ii) circle $x^{2}+y^{2}=a^{2}$;

(iii) hyperbola $x^{2}-y^{2}=a^{2}$.

We consider the area that lies above the $x$-axis, below the curve and between $x=0$ and $x=t$; namely