Unless specified, all approximating rectangles are assumed to have the same width.
Evaluate the upper and lower sums for
f(x) = 2 + sin(x),

0 ≤ x ≤ 𝜋
with n = 2, 4, and 8. (Round your answers to two decimal places.)
The x y-coordinate plane is given. A curve and two sets of 5 rectangles are graphed. The x-axis from x = a to x = b is divided into 5 subregions, each of which becomes a shared base for rectangles of width (b − a)/5.
The curve enters the window in the first quadrant above x = a, goes down and right, changes direction, goes up and right, changes direction, goes down and right, and exits the window in the first quadrant above x = b.
The first set of 5 rectangles extends up and intersects the curve at the higher value within each subregion.
The second set of 5 rectangles extends up and intersects the curve at the lower value within each subregion.

### Problem Restatement:
We are tasked with evaluating the upper and lower sums for the function:

\[
f(x) = 2 + \sin(x)
\]

over the interval \( 0 \leq x \leq \pi \), for \( n = 2, 4, \) and \( 8 \) subintervals. We will round the answers to two decimal places.

### Step 1: Understanding Upper and Lower Sums

For a function \( f(x) \), the **upper sum** and **lower sum** approximate the area under the curve using rectangles:
- **Upper sum**: The height of each rectangle is determined by the maximum value of \( f(x) \) in each subinterval.
- **Lower sum**: The height of each rectangle is determined by the minimum value of \( f(x) \) in each subinterval.

The width of each rectangle, for \( n \) rectangles, is:
\[
\Delta x = \frac{b - a}{n} = \frac{\pi - 0}{n} = \frac{\pi}{n}.
\]

### Step 2: Setting up the Subintervals

For the interval \( [0, \pi] \), we divide it into \( n \) equal subintervals, and the width \( \Delta x = \frac{\pi}{n} \).

#### Case 1: \( n = 2 \)
\[
\Delta x = \frac{\pi}{2}
\]
- Subintervals: \( [0, \frac{\pi}{2}], [\frac{\pi}{2}, \pi] \)

#### Case 2: \( n = 4 \)
\[
\Delta x = \frac{\pi}{4}
\]
- Subintervals: \( [0, \frac{\pi}{4}], [\frac{\pi}{4}, \frac{\pi}{2}], [\frac{\pi}{2}, \frac{3\pi}{4}], [\frac{3\pi}{4}, \pi] \)

#### Case 3: \( n = 8 \)
\[
\Delta x = \frac{\pi}{8}
\]
- Subintervals: \( [0, \frac{\pi}{8}], [\frac{\pi}{8}, \frac{2\pi}{8}], \dots, [\frac{7\pi}{8}, \pi] \)

### Step 3: Evaluate the Function \( f(x) = 2 + \sin(x) \) at Key Points

We will evaluate \( f(x) = 2 + \sin(x) \) at the endpoints and midpoints of each subinterval to determine the heights of the rectangles.

Let me now calculate the upper and lower sums for \( n = 2, 4, \) and \( 8 \).### Results for Upper and Lower Sums

For the function \( f(x) = 2 + \sin(x) \) over the interval \( [0, \pi] \), the upper and lower sums for \( n = 2, 4, 8 \) are:

- **For \( n = 2 \):**
  - Upper sum: 9.42
  - Lower sum: 6.28

- **For \( n = 4 \):**
  - Upper sum: 8.96
  - Lower sum: 7.39

- **For \( n = 8 \):**
  - Upper sum: 8.65
  - Lower sum: 7.86

These approximations get more accurate as \( n \) increases, with the upper and lower sums converging.

Would you like any further clarification or details on these calculations?

Here are 5 related questions:
1. How does increasing \( n \) affect the accuracy of the upper and lower sums?
2. How do the upper and lower sums compare to the exact integral of \( f(x) \)?
3. What happens if we calculate sums with \( n = 16 \) or higher?
4. How can the trapezoidal rule be used to approximate the integral of \( f(x) \)?
5. What is the relationship between the upper and lower sums and the Riemann integral?

**Tip:** As \( n \) increases, the difference between the upper and lower sums decreases, giving a better approximation of the actual area under the curve.

Unless specified, all approximating rectangles are assumed to have the same width. Evaluate the upper and lower sums for f(x) = 2 + sin(x), 0 ≤ x ≤ 𝜋 with n = 2, 4, and 8. (Round your answers to two decimal places.)

Explore how to calculate the upper and lower sums for the function f(x) = 2 + sin(x) over the interval [0, π], using n = 2, 4, and 8 subdivisions. This detailed guide walks through the steps, formulas, and approximations, offering key insights for calculus students.

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