Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit.

Consider the definite integral $\displaystyle \int_{4}^{8} x\sqrt{8 x - x^2}\, dx$
Then the most appropriate substitution to simplify this integral is $x = g(t)$ where
$g(t)$ =
Note: We are using $t$ as variable for angles instead of $\theta$, since there is no standard way to type $\theta$ on a computer keyboard.

After making this substitution and simplifying (using trig identities), we obtain the integral $\displaystyle \int_a^b f(t)\,dt$ where
$f(t)$ =
$a$ =
$b$ =
After evaluating this integral we obtain:
$\displaystyle \int_{4}^{8} x\sqrt{8 x - x^2}\, dx$ =