Find the area between y = 2 and y = cos x + 1 for 0 ≤ x ≤ 2π

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.

Learn how to calculate the area between the curves y = 2 and y = cos x + 1 over the interval 0 to 2π with a step-by-step solution. This problem involves using definite integrals to find the area under two curves. Suitable for advanced high school students and calculus learners.

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