Find the average value of f(x) = x^2 on [0, 3]

Problem: Find the average value of $f(x) = x^2$ on the interval $[0, 3]$.

Steps

1. Average value formula
   $$f_{\text{avg}} = \frac{1}{3 - 0} \int_{0}^{3} x^2 \, dx$$
2. Simplify the fraction
   $$f_{\text{avg}} = \frac{1}{3} \int_{0}^{3} x^2 \, dx$$
3. Integrate using the power rule
   $$f_{\text{avg}} = \frac{1}{3} \left[ \frac{x^3}{3} \right]_{0}^{3}$$
4. Evaluate at the upper and lower limits
   $$f_{\text{avg}} = \frac{1}{3} \left( \frac{3^3}{3} - \frac{0^3}{3} \right)$$
5. Simplify the expression
   $$f_{\text{avg}} = \frac{1}{3} \left( \frac{27}{3} \right) = \frac{1}{3} (9) = 3$$