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. $\frac{7}{3}$

Answer

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Answer

Final Answer: $\boxed{\frac{4}{3}}$

Steps

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: $\boxed{\frac{4}{3}}$