What is the area between the curve f(x)=x|x| and x-axis for x\in[-1,1]?
- A. \frac{2}{3} ✓
- B. \frac{1}{2}
- C. \frac{1}{4}
- D. \frac{1}{3}
Correct Answer: A. \frac{2}{3}
Explanation
The required area is A = \int_{-1}^1 |f(x)| \, dx. Since f(x) = x|x|, its absolute value is |x|x|| = x^2. Thus, the area is \int_{-1}^1 x^2 \, dx = \left[\frac{x^3}{3}\right]_{-1}^1 = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3}.
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