To solve this problem, we’ll go through each part step-by-step.

Part 1: Graph the Integrand

The integrand is given as: $f(x) = \frac{3}{x} + 3$ We are integrating over the interval $[1, 9]$.

Graph Selection: The correct graph should show the function $f(x) = \frac{3}{x} + 3$ over the interval $[1, 9]$. This function is decreasing as $x$ increases since $\frac{3}{x}$ diminishes while $3$ is constant. The graph should be a curve that starts higher at $x = 1$ and gradually decreases as $x$ moves to 9. Based on this, option C seems to match, with $f(x)$ passing through points close to $(1, 6)$ and $(9, 3.3)$.

Part 2: Calculate $\Delta x$ and the Grid Points

For a regular partition with $n = 4$, we divide the interval $[1, 9]$ into 4 equal subintervals.

1. Calculate $\Delta x$: $\Delta x = \frac{9 - 1}{4} = \frac{8}{4} = 2$

2. Determine the grid points: Starting at $x_0 = 1$ and adding $\Delta x = 2$ each time, we get:

   • $x_0 = 1$
   • $x_1 = 3$
   • $x_2 = 5$
   • $x_3 = 7$
   • $x_4 = 9$

So, $\Delta x = 2$ and the points are $x_0 = 1$, $x_1 = 3$, $x_2 = 5$, $x_3 = 7$, $x_4 = 9$.

Part 3: Left and Right Riemann Sums

The left and right Riemann sums are approximations of the area under the curve by rectangles whose heights are determined by the function's value at the left or right endpoints of each subinterval.

1. Left Riemann Sum: Using the left endpoints $x_0, x_1, x_2,$ and $x_3$: $L_n = f(x_0) \cdot \Delta x + f(x_1) \cdot \Delta x + f(x_2) \cdot \Delta x + f(x_3) \cdot \Delta x$ Calculating each term: $f(1) = \frac{3}{1} + 3 = 6$ $f(3) = \frac{3}{3} + 3 = 4$ $f(5) = \frac{3}{5} + 3 = 3.6$ $f(7) = \frac{3}{7} + 3 \approx 3.43$ Now, substitute into the left Riemann sum: $L_4 = (6)(2) + (4)(2) + (3.6)(2) + (3.43)(2)$ Simplifying: $L_4 = 12 + 8 + 7.2 + 6.86 = 34.06$

2. Right Riemann Sum: Using the right endpoints $x_1, x_2, x_3,$ and $x_4$: $R_n = f(x_1) \cdot \Delta x + f(x_2) \cdot \Delta x + f(x_3) \cdot \Delta x + f(x_4) \cdot \Delta x$ We have already calculated $f(x_1)$, $f(x_2)$, and $f(x_3)$ above. Now calculate $f(x_4)$: $f(9) = \frac{3}{9} + 3 = 3.33$ Substitute into the right Riemann sum: $R_4 = (4)(2) + (3.6)(2) + (3.43)(2) + (3.33)(2)$ Simplifying: $R_4 = 8 + 7.2 + 6.86 + 6.66 = 28.72$

Part 4: Determine Which Riemann Sum Underestimates or Overestimates

Since $f(x) = \frac{3}{x} + 3$ is a decreasing function over the interval $[1, 9]$:

• The left Riemann sum overestimates the actual area under the curve.
• The right Riemann sum underestimates the actual area.

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If you have questions or would like further explanation, let me know! Here are five related questions for further exploration:

1. How does increasing the value of $n$ affect the accuracy of the Riemann sums?
2. What happens to the left and right Riemann sums if $f(x)$ is increasing instead of decreasing?
3. How can the trapezoidal rule improve the accuracy of the Riemann sum approximations?
4. How would you set up a midpoint Riemann sum for this integral?
5. How does the actual integral value compare to the left and right Riemann sums?

Tip: For functions that are monotonic (always increasing or decreasing), the left and right Riemann sums will consistently overestimate or underestimate the integral value.