The integral \( \int_0^1 x^2 \, dx \) needs to be evaluated.
Step 1: Apply the power rule of integration
The power rule for integration states:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]
In this integral, \( n = 2 \). Applying the power rule yields:
\[
\int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1
\]
Step 2: Calculate the definite integral
Evaluate the expression at the upper and lower limits:
\[
= \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} - 0 = \frac{1}{3}
\]
Answer: The integral evaluates to \( \frac{1}{3} \). This corresponds to option (1).