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The problem asks to evaluate the integral:

\[
\int_{-5}^{5} (6 - |x|) \, dx
\]

by interpreting it in terms of areas. Let's break it down geometrically.

### Step 1: Understand the function \( 6 - |x| \)
The function \( 6 - |x| \) is a "V"-shaped function centered at \(x = 0\), with the vertex at \( (0, 6) \). The function decreases linearly on both sides of the y-axis, with slopes of \(-1\) and \(1\), making it symmetric about the y-axis. This is because the absolute value function \( |x| \) reflects negative \(x\)-values to the positive side.

### Step 2: Visualize the region
For the integral, we are interested in the area under the curve from \( x = -5 \) to \( x = 5 \).

- At \( x = -5 \), the value of the function is \( 6 - |-5| = 6 - 5 = 1 \).
- At \( x = 5 \), the value of the function is similarly \( 1 \).
- At \( x = 0 \), the function reaches its maximum value, which is \( 6 - |0| = 6 \).

Thus, the graph forms a symmetric trapezoid with a height of 6 at the center and a base extending from \( x = -5 \) to \( x = 5 \), which means the total base length is 10 units.

### Step 3: Divide the area into shapes
We can divide the region into two right triangles (on either side of the y-axis) and a central rectangle:

1. **Rectangle**: The width of the rectangle is 2 (from \( x = -1 \) to \( x = 1 \)), and the height is 6. 
   - Area of the rectangle = \( \text{width} \times \text{height} = 2 \times 6 = 12 \).

2. **Two right triangles**: Each triangle has a base of 4 (from \( x = -5 \) to \( x = -1 \), and similarly on the positive side), and a height of 5. 
   - Area of one triangle = \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 5 = 10 \).
   - Since there are two triangles, the total area for both = \( 2 \times 10 = 20 \).

### Step 4: Total area
Now, we add the areas of the rectangle and the two triangles:

\[
\text{Total area} = 12 + 20 = 32
\]

Thus, the value of the integral is:

\[
\int_{-5}^{5} (6 - |x|) \, dx = 32
\]

---

Would you like more details or have any questions? Here are some follow-up questions to deepen your understanding:

1. What is the general formula for integrating a piecewise function like \( 6 - |x| \)?
2. How would you approach this problem using algebraic methods instead of geometric interpretation?
3. How can we use symmetry in integrals to simplify problems like this?
4. How would this integral change if the limits of integration were different, say from \( -3 \) to \( 3 \)?
5. What happens to the area when the function is modified to something like \( 6 - x^2 \)?

**Tip**: Always check if a function is symmetric to make problems involving definite integrals easier to solve!

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 shapes from geometry. \(\int_{-5}^{5} (6 - |x|) dx\)

Evaluate the integral \(\int_{-5}^{5} (6 - |x|) dx\) by interpreting it geometrically. Learn how to break down the region into triangles and rectangles, and apply basic area formulas to solve the problem. This approach helps to understand the relationship between integrals and geometry.

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