Mistake 1: Forgetting to take absolute values. The integral from -1 to 1 of $x^3$ dx = 0, but the area between y = $x^3$ and the x-axis from -1 to 1 is 1/2. Mistake 2: Wrong limits of integration. Always solve for ALL intersection points, including those obtained by substitution. For y = $x^2$ and y = 2 - $x^2$, solving gives x = +/- 1, not just x = 1. Mistake 3: Integrating in the wrong variable. If the region has a vertical line segment at each x-value spanning from one curve to another, use dx. If horizontal, use dy. Mistake 4: Forgetting part of the boundary. The problem says "bounded by y = $x^2$, y = 0, and x = 2" — the region has THREE boundaries, and missing any one changes the area completely.