1. **State the problem:** Find the area under the curve $y=3x^2$ from $x=0$ to $x=b$ using a definite integral. 2. **Formula:** The area $A$ under a curve $y=f(x)$ from $x=a$ to $x=b$ is given by the definite integral: $$A=\int_a^b f(x)\,dx$$ 3. **Apply the formula:** Here, $f(x)=3x^2$, $a=0$, and $b$ is the upper limit. $$A=\int_0^b 3x^2\,dx$$ 4. **Integrate:** Use the power rule for integration: $$\int x^n dx=\frac{x^{n+1}}{n+1}+C$$ So, $$\int 3x^2 dx=3 \cdot \frac{x^{3}}{3}=x^3$$ 5. **Evaluate the definite integral:** $$A=\left[x^3\right]_0^b = b^3 - 0^3 = b^3$$ 6. **Interpretation:** The area under the curve $y=3x^2$ from $0$ to $b$ is $b^3$. **Final answer:** $$\boxed{b^3}$$