Step-by-Step Solution

1. First, find the points of intersection between the curves x^{2}=y and y=x+2. Set x^{2} = x+2, which gives x^{2} - x - 2 = 0. Factoring, we get (x-2)(x+1) = 0. Thus, x = 2 or x = -1. The corresponding y values are y = 4 and y = 1. So, the points of intersection are (2, 4) and (-1, 1).
2. Next, we need to find the area bounded by the curves and the x-axis. The region is divided into two parts. From x = -1 to x = 0, the area is under the line y = x+2. From x = 0 to x = 2, the area is bounded by the line y = x+2 above and the parabola y = x^{2} below.
3. Calculate the area from x = -1 to x = 0: Area_1 = \int_{-1}^{0} (x+2) dx = [\frac{x^{2}}{2} + 2x]_{-1}^{0} = (0) - (\frac{1}{2} - 2) = 2 - \frac{1}{2} = \frac{3}{2}.
4. Calculate the area from x = 0 to x = 2: Area_2 = \int_{0}^{2} (x+2 - x^{2}) dx = [\frac{x^{2}}{2} + 2x - \frac{x^{3}}{3}]_{0}^{2} = (\frac{4}{2} + 4 - \frac{8}{3}) - (0) = 2 + 4 - \frac{8}{3} = 6 - \frac{8}{3} = \frac{18 - 8}{3} = \frac{10}{3}.
5. The total area is the sum of Area_1 and Area_2: Total Area = Area_1 + Area_2 = \frac{3}{2} + \frac{10}{3} = \frac{9 + 20}{6} = \frac{29}{6}.