Finding cumulative distribution function, given density function using integration I am trying to find the cumulative distribution function given the following density function: $$f_X(x)=
\begin{cases}
\frac{2}{3\sqrt{x}}, &0\leq x < \frac{9}{16}\\
0, &\text{otherwise.}
\end{cases}
$$
I know I need to integrate $f_X(x)$ to find $F_X(x)$ but I am unsure what the initial value for this integral must be. Would this be correct integral, $\int\limits_{0}^{x}f_X(x)dx=\frac{4\sqrt{x}}{3}=F_X(x)$?

Yes, this is correct
$$
F_X(x) = P(X \le x) = \int_{-\infty}^x f_X(t)dt = \int_{0}^x f_X(t)dt = \frac{4}{2}\sqrt{x}, 
$$
for $x\in[0, 9/16)$, for $x \ge 9/16$, $F_X(x) = 1$, and for $x<0$, $F_X(x) = 0$.
Check
$$
\int_{0}^{9/16} f_X(x)dx = F_X(9/16) - F_X(0) = 1 - 0 =1.
$$
