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The image shows a graph of a piecewise function \( f(x) \), with a semi-circular portion from \( -a \) to \( a \), and two straight segments from \( a \) to \( 3a \). We need to evaluate the integral \( \int_{-a}^{3a} f(x) \, dx \) for \( a = 12 \), and round the final answer to 4 decimal places.

### Step 1: Break down the integral

The integral can be split into the following parts based on the graph:

\[
\int_{-a}^{3a} f(x) \, dx = \int_{-a}^{a} f(x) \, dx + \int_{a}^{2a} f(x) \, dx + \int_{2a}^{3a} f(x) \, dx
\]

### Step 2: Evaluate the semi-circle portion from \( -a \) to \( a \)

The graph shows that \( f(x) \) is a semi-circle on the interval \( [-a, a] \). The area under this semi-circle is given by the area of a circle of radius \( a \), which is \( \pi a^2 \), but since we only have a semi-circle, the area is:

\[
\text{Area of semi-circle} = \frac{1}{2} \pi a^2
\]

Given that the semi-circle is above the x-axis, this area is positive.

### Step 3: Evaluate the straight line portions from \( a \) to \( 3a \)

- From \( a \) to \( 2a \), the function is a straight line at \( -a \), so the integral is:

\[
\int_{a}^{2a} f(x) \, dx = (-a) \times (2a - a) = -a^2
\]

- From \( 2a \) to \( 3a \), the function is zero, so this integral contributes nothing:

\[
\int_{2a}^{3a} f(x) \, dx = 0
\]

### Step 4: Combine the results

Now, we combine the results from each interval:

\[
\int_{-a}^{3a} f(x) \, dx = \frac{1}{2} \pi a^2 + (-a^2) + 0
\]

Simplifying this expression:

\[
\int_{-a}^{3a} f(x) \, dx = \frac{1}{2} \pi a^2 - a^2
\]

### Step 5: Plug in \( a = 12 \)

Substituting \( a = 12 \) into the equation:

\[
\int_{-12}^{36} f(x) \, dx = \frac{1}{2} \pi (12)^2 - (12)^2
\]
\[
= \frac{1}{2} \pi \times 144 - 144
\]
\[
= 72\pi - 144
\]

### Step 6: Calculate the numerical value

Using \( \pi \approx 3.1416 \):

\[
72 \times 3.1416 = 226.1942
\]
\[
\int_{-12}^{36} f(x) \, dx \approx 226.1942 - 144 = 82.1942
\]

Thus, the value of the integral is approximately:

\[
\boxed{82.1942}
\]

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Do you want further details or have any questions?

Here are 5 related questions:
1. How do you calculate the area of a full circle?
2. What is the geometric interpretation of an integral?
3. Why is the integral of the straight-line portion negative?
4. How does the integral change if \( a \) takes on a different value?
5. What does it mean when a function has an integral equal to zero?

**Tip:** When splitting an integral, always pay attention to where the function crosses the x-axis, as this can change the sign of the integral.

If a = 12, then find the value of the integral ∫[from -a to 3a] f(x) dx based on the given graph.

This solution walks through the process of evaluating the integral of a piecewise function composed of a semi-circle and straight-line segments. It covers key concepts in integral calculus and geometry, including using the area formula for a semi-circle and handling piecewise-defined functions. Suitable for Grades 11-12.

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