No Integration required :)

The equation can be rewritten as $(|y|+2)^2+(x-4)^2=16$ . If $|y|$ is replaced with $y$ , the equation would be a circle centered at $(4,-2)$ . Because of the absolute value, the region of the circle below the $x$ -axis is replaced with the region of the graph above the $x$ -axis. In other words, the part of the graph above the $x$ -axis is reflected over the $x$ -axis, which means the total area is twice the area bound by the part of the graph above the $x$ -axis and the line $y=0$ .

We know that the radius is $4$ so the graph never enters the 2nd or 3rd quadrant. This means the area above the $x$ -axis is a circular cap with the base being the part of the $x$ -axis contained within the circle. The distance of the chord that represents the part of the $x$ -axis contained in the circle can be found by setting $y=0$ , meaning $x=4+2√3$ and $x=4-2√3$ so the distance of the chord would be $4√3$ .

We can draw radii from $(4,-2)$ to the endpoints of this chord and find the angle between the radii using the Law of Cosines since we know the radii are both $4$ . The angle comes out to be $120˙$ and the area of the circular cap would be $\frac{16π}{3}$ $-(0.5*4^2*0.5*\sin{120})$ and multiplying this by $2$ accounts for the reflection across the $x$ -axis. The final area comes out to be $\frac{32π}{3}$ $-8√3$ so $(a+b+c)=32+8+3=43$ which is the final answer.