Find the area under y = x^2 from 0 to 2

Problem: Find the area under $y = x^2$ from $0$ to $2$, represented as the definite integral $\int_{0}^{2} x^2 \, dx$.

Steps

1. Set up the definite integral
   $$\int_{0}^{2} x^2 \, dx$$
2. Find the antiderivative using the power rule
   $$\left[ \frac{x^3}{3} \right]_{0}^{2}$$
3. Evaluate the antiderivative at the upper and lower limits
   $$\left( \frac{2^3}{3} \right) - \left( \frac{0^3}{3} \right)$$
4. Simplify the expression
   $$\frac{8}{3} - 0 = \frac{8}{3}$$