Let's go through the integrals step-by-step.

From the image, we see the function $f(x)$ consists of two semicircles. The details of each semicircle are as follows:

1. First semicircle (below the x-axis):

   • Radius: 1 (from 0 to 2 on the x-axis).
   • Centered at $x = 1$.
   • This semicircle covers $x = 0$ to $x = 2$.
   • Area of the semicircle: $-\frac{1}{2} \pi \cdot (1^2) = -\frac{\pi}{2}$ (since it is below the x-axis, the area is negative).

2. Second semicircle (above the x-axis):

   • Radius: 1 (from 4 to 6 on the x-axis).
   • Centered at $x = 5$.
   • This semicircle covers $x = 4$ to $x = 6$.
   • Area of the semicircle: $\frac{1}{2} \pi \cdot (1^2) = \frac{\pi}{2}$.

Now, let's evaluate each integral based on this information.

Integral Solutions

a) $\int_0^2 3f(x) \, dx$

• The integral $\int_0^2 f(x) \, dx$ corresponds to the area of the first semicircle, which is $-\frac{\pi}{2}$.
• Thus, $\int_0^2 3f(x) \, dx = 3 \cdot \left(-\frac{\pi}{2}\right) = -\frac{3\pi}{2}$.

b) $\int_0^6 4f(x) \, dx$

• This integral covers both semicircles.
• The sum of the areas of the two semicircles is $-\frac{\pi}{2} + \frac{\pi}{2} = 0$.
• Therefore, $\int_0^6 4f(x) \, dx = 4 \cdot 0 = 0$.

c) $\int_1^4 5f(x) \, dx$

• This range includes part of the negative semicircle (from $x = 1$ to $x = 2$) and nothing for the second semicircle, as it starts at $x = 4$.
• The area from $x = 1$ to $x = 2$ is half of the negative semicircle area, or $-\frac{\pi}{4}$.
• Thus, $\int_1^4 5f(x) \, dx = 5 \cdot \left(-\frac{\pi}{4}\right) = -\frac{5\pi}{4}$.

d) $\int_1^6 |2f(x)| \, dx$

• Here, we take the absolute value of $f(x)$ and double it.
• The area from $x = 1$ to $x = 2$ (below the x-axis) is now positive $\frac{\pi}{4}$.
• The area from $x = 4$ to $x = 6$ (above the x-axis) remains $\frac{\pi}{2}$.
• So, $\int_1^6 |2f(x)| \, dx = 2 \cdot \left(\frac{\pi}{4} + \frac{\pi}{2}\right) = 2 \cdot \frac{3\pi}{4} = \frac{3\pi}{2}$.

Summary of Answers

1. $\int_0^2 3f(x) \, dx = -\frac{3\pi}{2}$
2. $\int_0^6 4f(x) \, dx = 0$
3. $\int_1^4 5f(x) \, dx = -\frac{5\pi}{4}$
4. $\int_1^6 |2f(x)| \, dx = \frac{3\pi}{2}$

Would you like further explanations or breakdowns for any part?

Additional Questions

1. What is the general approach to calculating integrals for functions that have distinct segments like semicircles?
2. How does the absolute value in an integral impact the final answer?
3. Can you explain why multiplying the function inside the integral by a constant changes the result proportionally?
4. What is the significance of taking the area below the x-axis as negative in integrals?
5. How would these integrals change if the semicircles had a different radius?

Tip

Always analyze the graph or function structure before integrating, as visual clues like symmetry or distinct sections can simplify your calculations.