Exponential dist expected value help

Q: The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula relates the cost C of completing this operation to the square of the time to completion. Find the mean and variance of C.

My work:

Since I need to solve for and first.

So, I started with finding ,

, by u-sub we have , so

and so, if that is correct, shouldn't and so, down to ?

Nevertheless, I keep ending up with a negative variance when I use this method, since my gets so large. Do I just have to crank out every integrand by way of integration by parts instead of using the gamma function?

Some direction would be great,

Thanks