<p>Evaluate <span>\(\int_{-3}^{3}\sqrt{x^2-1}\,dx\)</span></p>
Step-by-Step Solution
Key Concept: Identify where the expression under modulus changes sign and split the integral at those points.
<p><strong>Solution:</strong> The modulus function changes at points where $x^2-1=0$, i.e., at $x=\pm 1$.</p><p>So, $I = \int_{-3}^{-1}(x^2-1)\,dx + \int_{-1}^{1}(1-x^2)\,dx + \int_{1}^{3}(x^2-1)\,dx$</p><p>$= \left[\frac{x^3}{3}-x\right]_{-3}^{-1} + \left[x-\frac{x^3}{3}\right]_{-1}^{1} + \left[\frac{x^3}{3}-x\right]_{1}^{3}$</p><p>$= \frac{2}{3} + \frac{2}{3} + \frac{2}{3} + \frac{2}{3} + \frac{2}{3} + 6 + \frac{2}{3} = \frac{28}{3}$</p>
Correct Answer: 28/3