Calculating probability from a Probability Density Function

Question

Is it possible to calculate the probability of a continuous variable from a probability density function without integrating to find the CDF?

I have a PDF $$f(x) = \begin{cases}3x^2 & 0 \le x \le 1 \\ 0 & \text{otherwise}\end{cases}$$

I can calculate $P\left(X\le \dfrac{1}{2}\right)$ by integrating to find the cumulative density function, but I have been asked to calculate it without a CDF. Is this even possible?

Thanks

Answer 1

Thanks everyone. The answer was simpler than I thought. Nothing about the method. All about the representation of the CDF so you can ask questions like $P\left(X\le \dfrac{1}{2}\right)$.

The answer should be in the form:

$$F(x) = \begin{cases}0 & x \lt 0 \\x^3 & 0 \le x \le 1 \\ 1 & x \gt 1\end{cases}$$