I see I made an error in combining the areas from the two integrals. Let's correct that and calculate the total area properly.

### Step 3: Evaluate the Integrals

Let's re-evaluate each integral separately and then sum them correctly.

#### First Integral: From \( x = 2 \) to \( x = 3 \)

\[
\int_{2}^{3} \left( -\frac{3x}{2} + 3 \right) \, dx
\]

\[
= \left[ -\frac{3}{4}x^2 + 3x \right]_{2}^{3}
\]

\[
= \left( -\frac{3}{4}(3^2) + 3(3) \right) - \left( -\frac{3}{4}(2^2) + 3(2) \right)
\]

\[
= \left( -\frac{27}{4} + 9 \right) - \left( -\frac{12}{4} + 6 \right)
\]

\[
= \left( -\frac{27}{4} + \frac{36}{4} \right) - \left( -3 + 6 \right)
\]

\[
= \frac{9}{4} + 3 = \frac{21}{4} = 5.25
\]

#### Second Integral: From \( x = 3 \) to \( x = 6 \)

\[
\int_{3}^{6} \left( \frac{x}{2} - 3 \right) \, dx
\]

\[
= \left[ \frac{1}{4}x^2 - 3x \right]_{3}^{6}
\]

\[
= \left( \frac{1}{4}(6^2) - 3(6) \right) - \left( \frac{1}{4}(3^2) - 3(3) \right)
\]

\[
= \left( \frac{36}{4} - 18 \right) - \left( \frac{9}{4} - 9 \right)
\]

\[
= (9 - 18) - \left( \frac{9}{4} - 9 \right)
\]

\[
= -9 + \frac{36}{4} - \frac{9}{4} 
\]

\[
= -9 + \frac{27}{4} 
\]

\[
= -9 + \frac{36}{4} - \frac{9}{4} = \frac{-27}{4}  = \frac{27}{4}
\]

### Total Area Calculation

Now, we correctly add the areas from the two integrals:

\[
A_{\text{total}} = \frac{21}{4} + \frac{27}{4} = \frac{48}{4} = 12
\]

So, the correct total area of the region is \( \boxed{12} \).

Would you like more details on any part of the calculation or have any other questions?

---

**5 Related Questions:**

1. How do we handle the area calculation if the intersection points are not easily found?
2. How would you find the area between curves in polar coordinates?
3. What if the curves intersect at more than two points?
4. How can you verify the accuracy of your area calculations?
5. How do you calculate the area between curves when one function is a vertical line?

**Tip:** Double-check the integration limits and the integrand when computing the area between curves to avoid errors.

Learn how to calculate the total area using definite integrals with a detailed step-by-step guide. This problem involves evaluating integrals and summing areas correctly between given limits. Suitable for advanced high school students and calculus learners.

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