\(\int_{1}^{4}\displaystyle |x - 2|\,dx\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{5}{2}\)
The expression x − 2 changes sign at x = 2, so the integral is split at x = 2:
\(|x - 2| = 2 - x\) for \(1 \le x \le 2\)
\(|x - 2| = x - 2\) for \(2 \le x \le 4\)
\( \displaystyle \int_{1}^{4} |x - 2|\,dx = \int_{1}^{2} (2 - x)\,dx + \int_{2}^{4} (x - 2)\,dx \)
\( \displaystyle = \left[2x - \frac{x^2}{2}\right]_{1}^{2} + \left[\frac{x^2}{2} - 2x\right]_{2}^{4} \)
\( \displaystyle = \left(2 - \frac{3}{2}\right) + \left(0 - (-2)\right) \)
\( \displaystyle = \frac{1}{2} + 2 = \frac{5}{2} \)
Shortcut: The graph of |x − 2| on [1, 4] forms two triangles on the x-axis: base 1, height 1 (area 1/2) and base 2, height 2 (area 2). Total area = 5/2.