Find the definite integral of (2x – x^2) between the limits 2 and 1

To determine the definite integral of the function (2x – x^2) between the limits of 2 and 1, we first need to find the antiderivative of the function. By applying the power rule of integration, we can find that the antiderivative of 2x is x^2 and the antiderivative of x^2 is (1/3)x^3. Therefore, the antiderivative of (2x – x^2) is x^2 – (1/3)x^3.

Once we have found the antiderivative of the function, we can then evaluate the definite integral by substituting the upper limit (2) and the lower limit (1) into the antiderivative and subtracting the results. By substituting 2 into the antiderivative, we get 2^2 – (1/3)(2)^3 = 4 – (8/3) = 4/3. Similarly, by substituting 1 into the antiderivative, we get 1^2 – (1/3)(1)^3 = 1 – (1/3) = 2/3. 

Finally, to find the definite integral of (2x – x^2) between the limits of 2 and 1, we subtract the result of substituting the lower limit from the result of substituting the upper limit. Therefore, the definite integral of the function between the limits of 2 and 1 is 4/3 – 2/3 = 2/3.

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