Let $\mathrm{A}_1$ be the bounded area enclosed by the curves $y=x^2+2, x+y=8$ and $y$-axis that lies in the first quadrant. Let $\mathrm{A}_2$ be the bounded area enclosed by the curves $y=x^2+2, y^2=x, x=2$, and $y$-axis that lies in the first quadrant. Then $\mathrm{A}_1-\mathrm{A}_2$ is equal to

The area of the region enclosed between the circles $x^2+y^2=4$ and $x^2+(y-2)^2=4$ is:

The area of the region $\mathrm{A}=\left\{(x, y): 4 x^2+y^2 \leqslant 8\right.$ and $\left.y^2 \leqslant 4 x\right\}$ is:

Let the line $x=-1$ divide the area of the region $\left\{(x, y): 1+x^2 \leq y \leq 3-x\right\}$ in the ratio $m: n, \operatorname{gcd}(m, n)=1$. Then $m+n$ is equal to