The graph given in the image consists of three distinct regions:

1. A semicircle on the interval $[-a, a]$, which represents a positive area.
2. A line segment sloping downwards on the interval $[a, 2a]$, likely representing a negative area.
3. A flat horizontal line along the x-axis on the interval $[2a, 3a]$, contributing no area as $f(x) = 0$.

Given that $a = 11$, we need to compute the total area under the curve from $-a$ to $3a$ by dividing it into the respective parts and calculating the definite integral as the sum of these areas.

Step 1: Calculate the area of the semicircle

The equation of the semicircle has radius $a = 11$, and the area of a semicircle is given by: $A_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi a^2 = \frac{1}{2} \pi (11)^2 = \frac{1}{2} \pi \times 121$

Step 2: Calculate the area under the sloped line

The line on $[a, 2a]$ is a straight line sloping downward, representing a triangle. If we assume it is a linear function (constant slope), the area under the triangle can be computed using the area of a triangle formula: $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}$ Here, the base is $a = 11$, and the height is the negative value at $a$, which is $-a = -11$. So the area becomes: $A_{\text{triangle}} = \frac{1}{2} \times 11 \times (-11) = -60.5$

Step 3: Area from $2a$ to $3a$

Since $f(x) = 0$ in this region, the contribution of this area is zero.

Step 4: Total integral

Summing the areas of the semicircle and the triangle: $\int_{-a}^{3a} f(x)\, dx = A_{\text{semicircle}} + A_{\text{triangle}} = \frac{1}{2} \pi \times 121 + (-60.5)$ Let's calculate the numerical values.

I will now compute these values.The value of the integral $\int_{-a}^{3a} f(x) \, dx$, where $a = 11$, is approximately $129.5664$ when rounded to four decimal places.

Do you have any questions or would you like more details on any step?

Here are 5 related questions:

1. How is the definite integral related to the area under the curve?
2. Why does the area under the semicircle contribute positively to the integral?
3. How can you verify if the line segment on the interval $[a, 2a]$ forms a triangle?
4. What would change in the integral if the function for the line segment had a different slope?
5. How would you calculate the integral if the region from $2a$ to $3a$ was not zero?

Tip: When computing integrals geometrically, always check the shape and bounds to confirm how each region contributes (positively or negatively) to the total area.