To solve the integral

$\int_0^3 \left|2x - 9\right| \, dx,$

we will interpret it geometrically by analyzing the absolute value function. This involves understanding the shape of the curve and finding the area under it.

Step 1: Understand the behavior of the absolute value function

The function inside the absolute value is $2x - 9$. The absolute value function $|2x - 9|$ will be piecewise, meaning we need to break it into two regions based on the value of $2x - 9$.

First, find when $2x - 9 = 0$:

$2x - 9 = 0 \quad \Rightarrow \quad x = \frac{9}{2} = 4.5.$

This is the point where the expression $2x - 9$ changes sign. Therefore, the behavior of $|2x - 9|$ is:

• For $x < 4.5$, $2x - 9 < 0$, so $|2x - 9| = -(2x - 9) = 9 - 2x$.
• For $x > 4.5$, $2x - 9 > 0$, so $|2x - 9| = 2x - 9$.

Step 2: Break the integral into two parts

Since $x = 4.5$ lies outside the interval $[0, 3]$, we do not need to worry about the second case (for $x > 4.5$). Instead, the absolute value expression simplifies for all $x$ in $[0, 3]$ as:

$|2x - 9| = 9 - 2x.$

Thus, the integral becomes:

$\int_0^3 (9 - 2x) \, dx.$

Step 3: Evaluate the integral

Now, we can evaluate the integral:

$\int_0^3 (9 - 2x) \, dx.$

First, split the integral:

$\int_0^3 9 \, dx - \int_0^3 2x \, dx.$

Evaluate each part:

• The first integral is:

$\int_0^3 9 \, dx = 9x \Big|_0^3 = 9(3) - 9(0) = 27.$

• The second integral is:

$\int_0^3 2x \, dx = x^2 \Big|_0^3 = (3)^2 - (0)^2 = 9.$

Thus, the total area is:

$27 - 9 = 18.$

Final Answer

The value of the integral is:

$\boxed{18}.$

This represents the total area under the curve $|2x - 9|$ on the interval $[0, 3]$.