To solve the integral \(\int_{3}^{5} |x-4|\,dx\), we need to understand that the absolute value function \(|x-4|\) changes its behavior at \(x = 4\). Therefore, we should split the integral into two parts at this point and evaluate each part separately:
- For \(x \in [3, 4]\), \(|x-4| = -(x-4)\) because \(x-4\) is negative.
- For \(x \in [4, 5]\), \(|x-4| = x-4\) because \(x-4\) is positive.
Now, let's calculate the integral over each of these intervals separately:
- Evaluate \(\int_{3}^{4} -(x-4)\,dx\):
Rewriting the expression, we have:
- \(\int_{3}^{4} -(x-4)\,dx = \int_{3}^{4} -(x) + 4\,dx = \int_{3}^{4} (-x + 4)\,dx\)
This can be split into:
- \(\int_{3}^{4} (-x)\,dx + \int_{3}^{4} 4\,dx\)
Now compute these integrals:
- \(\int_{3}^{4} (-x)\,dx = -\left[\frac{x^2}{2}\right]_{3}^{4} = -\left[\frac{4^2}{2} - \frac{3^2}{2}\right] = -\left[\frac{16}{2} - \frac{9}{2}\right] = -\left[\frac{7}{2}\right] = -\frac{7}{2}\)
- \(\int_{3}^{4} 4\,dx = 4[x]_{3}^{4} = 4(4 - 3) = 4(1) = 4\)
Combine these results:
- \(\int_{3}^{4} (-x + 4)\,dx = -\frac{7}{2} + 4 = \frac{8}{2} - \frac{7}{2} = \frac{1}{2}\)
- Evaluate \(\int_{4}^{5} (x-4)\,dx\):
- \(\int_{4}^{5} (x-4)\,dx = \int_{4}^{5} x\,dx - \int_{4}^{5} 4\,dx\)
Now compute these integrals:
- \(\int_{4}^{5} x\,dx = \left[\frac{x^2}{2}\right]_{4}^{5} = \left[\frac{5^2}{2} - \frac{4^2}{2}\right] = \left[\frac{25}{2} - \frac{16}{2}\right] = \frac{9}{2}\)
- \(\int_{4}^{5} 4\,dx = 4[x]_{4}^{5} = 4(5 - 4) = 4(1) = 4\)
Combine these results:
- \(\int_{4}^{5} (x-4)\,dx = \frac{9}{2} - 4 = \frac{9}{2} - \frac{8}{2} = \frac{1}{2}\)
Final Calculation: Add the results of the two integrals:
- \(\int_{3}^{5} |x-4|\,dx = \int_{3}^{4} -(x-4)\,dx + \int_{4}^{5} (x-4)\,dx = \frac{1}{2} + \frac{1}{2} = 1\)
Therefore, the evaluated integral \(\int_{3}^{5} |x-4|\,dx\) is \(1\). Thus, the correct answer is \(1\).