Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using geometry.

$$ \int_0^3\!\left\vert2x-9\right\vert\,\mathrm{d}x $$

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] $.

Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using geometry. $$ \int_0^3\!\left\vert2x-9\right\vert\,\mathrm{d}x $$

Learn how to evaluate the integral $$ \int_0^3\!\left\vert2x-9\right\vert\,\mathrm{d}x $$ by interpreting it in terms of areas. This solution involves understanding the absolute value function, breaking it into piecewise linear parts, and calculating the enclosed area using geometry. Suitable for high school and college calculus students.

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