1. The problem is to calculate the definite integral of the function $x^2$ from 0 to 100. 2. The formula for the definite integral of a function $f(x)$ from $a$ to $b$ is: $$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F(x)$ is the antiderivative of $f(x)$. 3. For $f(x) = x^2$, the antiderivative is: $$F(x) = \frac{x^3}{3}$$ 4. Evaluate $F(x)$ at the bounds 0 and 100: $$F(100) = \frac{100^3}{3} = \frac{1,000,000}{3}$$ $$F(0) = \frac{0^3}{3} = 0$$ 5. Calculate the definite integral: $$\int_0^{100} x^2 \, dx = F(100) - F(0) = \frac{1,000,000}{3} - 0 = \frac{1,000,000}{3}$$ 6. The final answer is: $$\boxed{\frac{1,000,000}{3}}$$