Let us carefully analyze and solve the given problem step by step.

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Observations:

1. The graph consists of two semicircles:

   • A lower semicircle between $x = 0$ and $x = 2$, centered at $x = 1$ with radius $1$, below the $x$-axis.
   • An upper semicircle between $x = 2$ and $x = 6$, centered at $x = 4$ with radius $2$, above the $x$-axis.

2. The integrals involve different constants scaling $f(x)$ and absolute values.

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Step 1: Key Calculations

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

• The interval $x \in [0, 2]$ covers the lower semicircle.
• The area of a semicircle is: $A = \frac{1}{2} \pi r^2$ where $r = 1$ (radius of the semicircle). Hence, the area is: $A = \frac{1}{2} \pi (1)^2 = \frac{\pi}{2}.$ Since the semicircle is below the $x$-axis, the integral of $f(x)$ over this interval is: $\int_0^2 f(x) \, dx = -\frac{\pi}{2}.$ Multiplying by 2 (as per the problem): $\int_0^2 2f(x) \, dx = 2 \times \left(-\frac{\pi}{2}\right) = -\pi.$

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

• The interval $x \in [0, 6]$ includes both the lower and upper semicircles.
• The integral of $f(x)$ for the lower semicircle (from $x = 0$ to $x = 2$) is $-\frac{\pi}{2}$.
• The integral of $f(x)$ for the upper semicircle (from $x = 2$ to $x = 6$) is: $\int_2^6 f(x) \, dx = \frac{1}{2} \pi (2)^2 = 2\pi.$ Adding these together: $\int_0^6 f(x) \, dx = -\frac{\pi}{2} + 2\pi = \frac{3\pi}{2}.$ Multiplying by 4: $\int_0^6 4f(x) \, dx = 4 \times \frac{3\pi}{2} = 6\pi.$

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

• The interval $x \in [1, 4]$ includes:
  • The right half of the lower semicircle ($x = 1$ to $x = 2$): \text{Area} = \frac{1}{4} \pi (1)^2 = \frac{\pi}{4}, \text{ negative (below x-axis).}
  • The left half of the upper semicircle ($x = 2$ to $x = 4$): \text{Area} = \frac{1}{4} \pi (2)^2 = \pi, \text{ positive (above x-axis).} Adding these: $\int_1^4 f(x) \, dx = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}.$ Multiplying by 5: $\int_1^4 5f(x) \, dx = 5 \times \frac{3\pi}{4} = \frac{15\pi}{4}.$

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

• The absolute value of $f(x)$ turns both semicircles' contributions positive:
  • Lower semicircle ($x = 0$ to $x = 2$): $\frac{\pi}{2}$.
  • Upper semicircle ($x = 2$ to $x = 6$): $2\pi$. Adding these together: $\int_0^6 |f(x)| \, dx = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2}.$ Multiplying by 4: $\int_0^6 |4f(x)| \, dx = 4 \times \frac{5\pi}{2} = 10\pi.$

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Final Answers:

a) $-\pi$
b) $6\pi$
c) $\frac{15\pi}{4}$
d) $10\pi$

Would you like a detailed explanation for any step?

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Related Questions:

1. How does the radius of a semicircle affect its integral?
2. Why does the sign of the area depend on the position relative to the $x$-axis?
3. How do absolute value integrals affect the result of a definite integral?
4. What is the geometric significance of scaling a function like $2f(x)$ or $4f(x)$?
5. How can symmetry be used to simplify integral calculations?

Tip:

Always sketch or analyze the geometry of a graph for integrals involving piecewise or geometric functions. This can simplify the computation significantly!