For the last part, why is it (right - left)? It is because the Method 2, setting the limit section on y-axis and evaluating the change of x-axis. That means (y+2)-(y^2) = dx.   From the x-axis perspective, the x value from straight line (x=y+2) is definitely greater than the  x value from Arc BC (x=y^2). Or you can just simply draw a line e.g. y=1(horizontal line).  You will know the x value from straight line> x value from Arc BC####[]