Find the exact area under the curve between the indicated values of $x$. $y=2 x-x^{2} ;$ between $x=0$ and $x=2$ A. $\frac{5}{3}$ B. $\frac{2}{3}$ C. $\frac{4}{3}$ D. $\frac{7}{3}$

Step 1 ：The problem is to find the exact area under the curve of the function \(y=2x - x^2\) between \(x=0\) and \(x=2\).

Step 2 ：The area under the curve of a function between two points can be found by integrating the function from the lower limit to the upper limit.

Step 3 ：So, we need to integrate the function \(y=2x - x^2\) from \(x=0\) to \(x=2\).

Step 4 ：The result of the integration is \(\frac{4}{3}\), which is the exact area under the curve between \(x=0\) and \(x=2\).

Step 5 ：Final Answer: \(\boxed{\frac{4}{3}}\)