To solve the problem, let's break it down step by step:

1. Understanding the Graph

• The graph depicts two semicircles:
  • The left semicircle is centered at $x = 0$ and has a radius of 1.
  • The right semicircle is centered at $x = 4$ and has a radius of 2.
• The integrals involve combinations of these semicircles. The integral of $f(x)$ corresponds to the signed area under the graph.

2. Calculate Key Areas

Left Semicircle (radius $r = 1$):

• Area = $\frac{1}{2} \pi r^2 = \frac{1}{2} \pi (1)^2 = \frac{\pi}{2}$.
• This area is below the x-axis, so its contribution to the integral is $-\frac{\pi}{2}$.

Right Semicircle (radius $r = 2$):

• Area = $\frac{1}{2} \pi r^2 = \frac{1}{2} \pi (2)^2 = 2\pi$.
• This area is above the x-axis, so its contribution to the integral is $+2\pi$.

3. Evaluate the Integrals

Part (a): $\int_0^6 2f(x) \, dx$

• Multiply each area by 2:
  • Left semicircle: $2(-\frac{\pi}{2}) = -\pi$,
  • Right semicircle: $2(2\pi) = 4\pi$.
• Total: $-\pi + 4\pi = 3\pi$.

Part (b): $\int_0^6 3f(x) \, dx$

• Multiply each area by 3:
  • Left semicircle: $3(-\frac{\pi}{2}) = -\frac{3\pi}{2}$,
  • Right semicircle: $3(2\pi) = 6\pi$.
• Total: $-\frac{3\pi}{2} + 6\pi = \frac{9\pi}{2}$.

Part (c): $\int_0^4 5f(x) \, dx$

• From $x = 0$ to $x = 4$, only the left semicircle contributes:
  • Left semicircle: $5(-\frac{\pi}{2}) = -\frac{5\pi}{2}$.
• Total: $-\frac{5\pi}{2}$.

Part (d): $\int_1^6 3f(x) \, dx$

• From $x = 1$ to $x = 6$, only part of the left semicircle (from $x = 1$ to $x = 0$) and the full right semicircle contribute.
• Area of left semicircle from $x = 1$ to $x = 0$:
  • This is one-fourth of the semicircle, so its area is $-\frac{1}{4} \cdot \frac{\pi}{2} = -\frac{\pi}{8}$.
  • Scaled by 3: $3(-\frac{\pi}{8}) = -\frac{3\pi}{8}$.
• Right semicircle scaled by 3: $3(2\pi) = 6\pi$.
• Total: $-\frac{3\pi}{8} + 6\pi = \frac{45\pi}{8}$.

4. Final Answers

• (a) $3\pi$,
• (b) $\frac{9\pi}{2}$,
• (c) $-\frac{5\pi}{2}$,
• (d) $\frac{45\pi}{8}$.

Let me know if you'd like additional explanations or help!

Related Questions:

1. How is the signed area of a semicircle calculated?
2. What is the significance of scaling $f(x)$ by a constant before integration?
3. How would the integral change if the left semicircle was above the x-axis?
4. What is the integral from $x = 0$ to $x = 1$ for this graph?
5. How does symmetry help simplify definite integrals for periodic functions?

Tip:

Always consider the sign of the area when dealing with definite integrals—negative for regions below the x-axis and positive for regions above it.