To calculate the total area enclosed by the man's path, we divide the area into three distinct regions and calculate each separately:

1. Region A: The area under the curve $f(x) = \sin\left(\frac{\pi x}{2}\right) + 2$ from $x = 0$ to $x = 2$.
2. Region B: The area of the vertical segment connecting $(2,2)$ to $(2,0)$.
3. Region C: The area of the rectangle from $(2,0)$ to $(0,0)$.

Step 1: Area under $f(x)$

The function is $f(x) = \sin\left(\frac{\pi x}{2}\right) + 2$. We integrate $f(x)$ from $x = 0$ to $x = 2$:

$\text{Area under the curve} = \int_0^2 \left(\sin\left(\frac{\pi x}{2}\right) + 2\right) \, dx.$

Break the integral into two parts:

$\int_0^2 \sin\left(\frac{\pi x}{2}\right) dx + \int_0^2 2 \, dx.$

1. First term: Solve $\int_0^2 \sin\left(\frac{\pi x}{2}\right) dx$: Let $u = \frac{\pi x}{2}$, so $du = \frac{\pi}{2} dx$, or $dx = \frac{2}{\pi} du$. Change the limits:

   • When $x = 0$, $u = 0$.
   • When $x = 2$, $u = \pi$.

   The integral becomes: $\int_0^2 \sin\left(\frac{\pi x}{2}\right) dx = \frac{2}{\pi} \int_0^\pi \sin(u) \, du.$

   The integral of $\sin(u)$ is $-\cos(u)$: [ \frac{2}{\pi} \int_0^\pi \sin(u) , du = \frac{2}{\pi} \left[ -\cos(u) \right]_0^\pi = \frac{2}{\pi} \left[ -\cos(\pi) + \cos(0) \right]. ]

   Using $\cos(\pi) = -1$ and $\cos(0) = 1$: $\frac{2}{\pi} \left[ -(-1) + 1 \right] = \frac{2}{\pi} (1 + 1) = \frac{4}{\pi}.$

2. Second term: Solve $\int_0^2 2 \, dx$: $\int_0^2 2 \, dx = 2x \big|_0^2 = 2(2) - 2(0) = 4.$

Adding the two parts: $\text{Area under the curve} = \frac{4}{\pi} + 4.$

Step 2: Area of the rectangle (Region C)

The rectangle has a width of $2$ and a height of $2$. Its area is: $\text{Area of the rectangle} = 2 \times 2 = 4.$

Step 3: Total enclosed area

The total enclosed area is the sum of the areas from Steps 1 and 2: $\text{Total area} = \left(\frac{4}{\pi} + 4\right) + 4 = \frac{4}{\pi} + 8.$

Final Answer:

The total area contained by the path is: $\boxed{\frac{4}{\pi} + 8 \, \text{square miles}.}$

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Would you like further clarification on any step? Here are related questions to explore:

1. How is the integral of trigonometric functions calculated?
2. Why does the rectangle contribute to the total area?
3. How do limits of integration affect area calculations?
4. Can this problem be solved geometrically without integration?
5. What if the curve was a different trigonometric function?

Tip: Always double-check your limits and substitutions when working with definite integrals!