<p>By drawing the graphs, find \(f(x)\) as shown and evaluate \[\int_{-1}^{1} f(x)\, dx\] where \(f(x)\) is the piecewise linear function forming triangles with peak value \(1/2\) at \(x = 0\), \(x = \pm 1\) as shown in the graph.</p>
Step-by-Step Solution
Key Concept: Recognize that the piecewise linear function forms two symmetric triangles with vertices at (-1,0), (0,1/2), and (1,0). Use geometry to find the area rather than integration, leveraging symmetry.
<p><strong>Step 1:</strong> Identify the piecewise linear function structure. The graph shows:</p><ul><li>For $x \in [-1, 0]$: line from $(-1, 0)$ to $(0, 1/2)$, so $f(x) = \frac{1}{2}(x+1)$</li><li>For $x \in [0, 1]$: line from $(0, 1/2)$ to $(1, 0)$, so $f(x) = \frac{1}{2}(1-x)$</li></ul><p><strong>Step 2:</strong> Recognize the region under the curve forms a single triangle with:</p><ul><li>Base = 2 (from $x = -1$ to $x = 1$)</li><li>Height = $\frac{1}{2}$ (peak at $x = 0$)</li></ul><p><strong>Step 3:</strong> Apply the geometric formula for triangle area:</p><p>$$\int_{-1}^{1} f(x)\,dx = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times \frac{1}{2} = \frac{1}{2}$$</p><p>∴ Answer: <strong>0.5</strong></p>
Correct Answer: 0.5