Let's go through each part of the problem based on the image.

The function given is: $f(x) = \frac{5}{2 + x^2}$ and we are tasked with estimating the area under the curve from $x = 3$ to $x = 5$ using a Riemann sum with left endpoints and four equal-width rectangles.

Part (a): Determine the Endpoints

The interval $[3, 5]$ is divided into four subintervals of equal width.

1. Width of each subinterval: $\Delta x = \frac{5 - 3}{4} = \frac{2}{4} = 0.5$

2. Left endpoints: Since we are using left endpoints, we begin at $x = 3$ and add $\Delta x = 0.5$ for each subsequent endpoint.

   The left endpoints are: $x_0 = 3, \quad x_1 = 3.5, \quad x_2 = 4, \quad x_3 = 4.5$

Part (b): Calculate $f(x)$ at Each Endpoint

Now, we evaluate $f(x)$ at each of these left endpoints. Each calculation should be rounded to three decimal places.

1. $f(x_0) = f(3) = \frac{5}{2 + 3^2} = \frac{5}{2 + 9} = \frac{5}{11} \approx 0.455$
2. $f(x_1) = f(3.5) = \frac{5}{2 + (3.5)^2} = \frac{5}{2 + 12.25} = \frac{5}{14.25} \approx 0.351$
3. $f(x_2) = f(4) = \frac{5}{2 + 4^2} = \frac{5}{2 + 16} = \frac{5}{18} \approx 0.278$
4. $f(x_3) = f(4.5) = \frac{5}{2 + (4.5)^2} = \frac{5}{2 + 20.25} = \frac{5}{22.25} \approx 0.225$

Part (c): Calculate the Riemann Sum

Using the left endpoints and four intervals of equal length, the Riemann sum is calculated as follows:

$R = \Delta x \left( f(x_0) + f(x_1) + f(x_2) + f(x_3) \right)$

Substitute the values we found:

$R = 0.5 \left( 0.455 + 0.351 + 0.278 + 0.225 \right)$

Calculate the sum of the values inside the parentheses:

$R = 0.5 \times (1.309) = 0.6545$

Rounded to three decimal places:

$R \approx 0.655$

Summary of Answers

• (a) Left endpoints: $x_0 = 3$, $x_1 = 3.5$, $x_2 = 4$, $x_3 = 4.5$
• (b) Values of $f(x)$:
  • $f(x_0) \approx 0.455$
  • $f(x_1) \approx 0.351$
  • $f(x_2) \approx 0.278$
  • $f(x_3) \approx 0.225$
• (c) Riemann sum: $R \approx 0.655$

Would you like further details on any step?

Related Questions:

1. How would the answer change if we used right endpoints instead of left endpoints?
2. How does the Riemann sum compare to the exact integral for this function over the interval $[3, 5]$?
3. What would be the effect of increasing the number of rectangles on the accuracy of the Riemann sum?
4. How is the midpoint Riemann sum different from the left or right Riemann sum in terms of accuracy?
5. Could we approximate this area using the trapezoidal rule instead? If so, how?

Tip:

When working with Riemann sums, using more intervals or smaller widths generally leads to a more accurate approximation of the integral.