Questions: Find the requested approximation for the definite integral using the indicated value of n. Give all answers to three decimal places, when necessary. Find Ln, Rn, and their average ∫ from 3 to 6 (x^2 - 2) dx: n=3

Solution Steps

To approximate the definite integral using the left endpoint (Ln) and right endpoint (Rn) methods, we first divide the interval [3, 6] into n subintervals. For each subinterval, we calculate the function value at the left endpoint for Ln and at the right endpoint for Rn. We then multiply these values by the width of the subintervals and sum them up to get Ln and Rn, respectively. Finally, we calculate the average of Ln and Rn.

We start with the function \( f(x) = x^2 - 2 \) that we need to integrate over the interval \([3, 6]\).

The interval \([3, 6]\) is divided into \( n = 3 \) subintervals. The width of each subinterval is calculated as: \[ \Delta x = \frac{b - a}{n} = \frac{6 - 3}{3} = 1.0 \]

Using the left endpoints of the subintervals, we compute \( L_n \): \[ L_n = \sum_{i=0}^{n-1} f(a + i \Delta x) \Delta x \] Calculating the function values:

• \( f(3) = 3^2 - 2 = 7 \)
• \( f(4) = 4^2 - 2 = 14 \)
• \( f(5) = 5^2 - 2 = 23 \)

Thus, \[ L_n = (f(3) + f(4) + f(5)) \Delta x = (7 + 14 + 23) \cdot 1.0 = 44.0 \]

Using the right endpoints of the subintervals, we compute \( R_n \): \[ R_n = \sum_{i=1}^{n} f(a + i \Delta x) \Delta x \] Calculating the function values:

• \( f(4) = 14 \)
• \( f(5) = 23 \)
• \( f(6) = 6^2 - 2 = 34 \)

Thus, \[ R_n = (f(4) + f(5) + f(6)) \Delta x = (14 + 23 + 34) \cdot 1.0 = 71.0 \]

The average of the left and right endpoint approximations is given by: \[ \text{Average} = \frac{L_n + R_n}{2} = \frac{44.0 + 71.0}{2} = 57.5 \]

Final Answer