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.