What is \int_{0}^{2}f(x)\,dx equal to?
Direction: Consider the following for the two (02) items that follow : Let f(x)=|x^{2}-x-2|.
- A. 0
- B. 1
- C. 5/3
- D. 10/3 ✓
Correct Answer: D. 10/3
Explanation
The roots of x^2-x-2 = (x-2)(x+1) are 2 and -1. In the interval , the expression is non-positive, so f(x) = -(x^2-x-2) = 2+x-x^2. Integrating this from 0 to 2: \int_0^2 (2+x-x^2)\,dx = [2x + \frac{x^2}{2} - \frac{x^3}{3}]_0^2 = 4 + 2 - \frac{8}{3} = 6 - \frac{8}{3} = \frac{10}{3}.
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