The value of \(\int_{1}^{4}\displaystyle |x - 1|\,dx\) is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{9}{2}\)
For \(1 \le x \le 4\), \(x - 1 \ge 0\), so the modulus can simply be dropped: \(|x - 1| = x - 1\).
\( \displaystyle \int_{1}^{4} |x - 1|\,dx = \int_{1}^{4} (x - 1)\,dx \)
\( \displaystyle = \left[\frac{x^2}{2} - x\right]_{1}^{4} \)
\( \displaystyle = (8 - 4) - \left(\frac{1}{2} - 1\right) \)
\( \displaystyle = 4 + \frac{1}{2} = \frac{9}{2} \)
Shortcut: The graph of y = x − 1 from x = 1 to x = 4 encloses with the x-axis a right triangle with base 3 and height 3, of area \(\frac{1}{2} \times 3 \times 3 = \frac{9}{2}\).