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Solve the following.
Evaluate: ∫₄⁵ |x - 5| dx.
Evaluate: $\int_{4}^{5} |x - 5| \, dx$.
Answer
Answer
Integral: $\int_{4}^{5} \left(5 - x\right) dx$
For $4 \le x \le 5$, $x - 5 \le 0$, so $|x-5| = 5 - x$.
$\int_4^5 (5-x)\,dx = \left[5x - \frac{x^2}{2}\right]_4^5 = \left(25 - \frac{25}{2}\right) - (20 - 8) = \frac{25}{2} - 12 = \frac{1}{2}$.
Final answer: $\frac{1}{2}$