To solve the integral $\int_{3}^{6} |3x - 15| \, dx$, follow these steps:

Step-by-Step Solution:

1. Understand the nature of the absolute value function:

   • The expression inside the absolute value, $3x - 15$, can be positive or negative depending on the value of $x$.
   • Solve $3x - 15 = 0$ to find the critical point where the expression changes sign: $3x = 15 \implies x = 5.$
   • For $x < 5$, $3x - 15 < 0$, so $|3x - 15| = -(3x - 15)$.
   • For $x \geq 5$, $3x - 15 \geq 0$, so $|3x - 15| = 3x - 15$.

2. Split the integral at the critical point $x = 5$: Since the integral spans the interval $[3, 6]$, and the critical point $x = 5$ lies in this interval, split the integral into two parts: $\int_{3}^{6} |3x - 15| \, dx = \int_{3}^{5} -(3x - 15) \, dx + \int_{5}^{6} (3x - 15) \, dx.$

3. Evaluate each integral:

   • For the first part: $\int_{3}^{5} -(3x - 15) \, dx = \int_{3}^{5} (-3x + 15) \, dx.$ Compute the antiderivative: $\int (-3x + 15) \, dx = -\frac{3x^2}{2} + 15x.$ Evaluate this from $x = 3$ to $x = 5$: $\left[ -\frac{3x^2}{2} + 15x \right]_{3}^{5} = \left(-\frac{3(5)^2}{2} + 15(5)\right) - \left(-\frac{3(3)^2}{2} + 15(3)\right).$ Simplify: $= \left(-\frac{75}{2} + 75\right) - \left(-\frac{27}{2} + 45\right)$ $= \frac{75}{2} - \frac{27}{2} = \frac{48}{2} = 24.$

   • For the second part: $\int_{5}^{6} (3x - 15) \, dx.$ Compute the antiderivative: $\int (3x - 15) \, dx = \frac{3x^2}{2} - 15x.$ Evaluate this from $x = 5$ to $x = 6$: $\left[ \frac{3x^2}{2} - 15x \right]_{5}^{6} = \left(\frac{3(6)^2}{2} - 15(6)\right) - \left(\frac{3(5)^2}{2} - 15(5)\right).$ Simplify: $= \left(\frac{108}{2} - 90\right) - \left(\frac{75}{2} - 75\right)$ $= 54 - 90 + 75 - \frac{75}{2}$ [ = \frac{27 /