Find $E(X^2)$ for this distribution

Question

A random variable $X$ has the PDF

$$f_X(x)=\begin{cases} Cx^2 & -1\le x\le 2,\\ 0 & \text{otherwise}, \end{cases}$$ and $Y=X^2$ . Calculate $E(Y)$.

Answer 1

If $Y=g(X)$, then $$\mathbb E[Y]=\int_{\mathbb R}g(x)f_X(x)dx.$$

Answer 2

Since $C^{-1}=\int_{-1}^2x^2dx=\frac{2^3-(-1)^3}{3}=3$, $\Bbb EX^2=\frac13\int_{-1}^2x^4dx=\frac{2^4-(-1)^4}{15}=1$.