Expected value of area of a triangle 
Two (possibly equal) integers $x$ and $y$ are chosen such that $1 \le x \le 5$ and $1 \le y \le 5$. What is the expected value of the area of the triangle with vertices $(0, 0)$, $(x, 0)$, and $(0, y)$?

I ended up getting$${1\over2}\left({{1+5}\over2}\right)\left({{1+5}\over2}\right)= {9\over2}$$Is this correct?

I assume, you have two independent random variables $x$ and $y$ with uniform distribution on the integers between 1 and 5. Each integer has a probability of $1/5$, so the probability of the pair $(x,y)$ is
$$p(x,y) = \frac{1}{25}$$
The Triangle size $  A(x,y) = \frac{xy}{2}$ is also a random variable. Its expectation value is
\begin{align}
\mathbb{E}(A) &= \sum_{x,y = 1}^5 A(x,y) p(x,y) = \sum_{x,y = 1}^5 \frac{xy}{2} \frac{1}{25} = \frac{1}{50} \left( \sum_{x = 1}^5 x \right) \left( \sum_{y = 1}^5 y \right) = \frac{15^2}{50} = \frac{9}{2}
\end{align}
So your result is apparently correct!
