To find the area between the curves $y = 2$ and $y = \cos x + 1$ over the interval $0 \leq x \leq 2\pi$, we calculate the definite integral of the difference between the two functions over that interval.

Step 1: Set Up the Integral

The area $A$ between the curves is given by:

$A = \int_{0}^{2\pi} \left[2 - (\cos x + 1)\right] \, dx$

Step 2: Simplify the Integrand

Simplifying the expression inside the integral:

$A = \int_{0}^{2\pi} \left[2 - \cos x - 1\right] \, dx = \int_{0}^{2\pi} \left(1 - \cos x\right) \, dx$

Step 3: Integrate

Now, integrate the simplified expression:

$A = \int_{0}^{2\pi} 1 \, dx - \int_{0}^{2\pi} \cos x \, dx$

1. The first integral is straightforward: $\int_{0}^{2\pi} 1 \, dx = \left. x \right|_{0}^{2\pi} = 2\pi$

2. The second integral is: $\int_{0}^{2\pi} \cos x \, dx = \left. \sin x \right|_{0}^{2\pi} = \sin(2\pi) - \sin(0) = 0 - 0 = 0$

Step 4: Calculate the Area

Subtract the results of the two integrals:

$A = 2\pi - 0 = 2\pi$

Final Answer

The area between the curves $y = 2$ and $y = \cos x + 1$ from $0 \leq x \leq 2\pi$ is $2\pi$ square units.

Would you like more details on any step or have any further questions?

Here are some related questions to explore:

1. How would the area change if the interval was $0 \leq x \leq \pi$?
2. What is the area between the curves $y = \cos x + 1$ and $y = 1$ over the same interval?
3. How would you find the area between $y = \cos x + 1$ and $y = -\cos x + 1$?
4. What happens to the area if you shift the curve $y = \cos x + 1$ vertically by 1 unit?
5. How would the integration limits change if the region were confined to $y = 1.5$?

Tip: When dealing with areas between curves, always ensure the integrand represents the correct difference between the upper and lower functions.