X is a continuous random variable with probability density function $f(x)$. The density is the curve $y=x^{2}$ for $0\le x\le 1$, followed by the straight line running from the point $(1;\,1)$ down to the point $(h;\,0)$ on the x-axis, and $f(x)=0$ elsewhere.

The total area under a probability density function is 1.

The curved piece contributes $\displaystyle\int_{0}^{1}x^{2}\,dx=\left[\frac{x^{3}}{3}\right]_{0}^{1}=\frac{1}{3}$.

The straight piece is a triangle of base $h-1$ and height 1, so it contributes $\frac{1}{2}(h-1)$.

$\dfrac{1}{3}+\dfrac{h-1}{2}=1$, so $\dfrac{h-1}{2}=\dfrac{2}{3}$, giving $h-1=\dfrac{4}{3}$ and $h=\dfrac{7}{3}=2,33$.