Questions: Find Ln, Rn, and their average for the definite integral below using the indicated value of n. ∫ from 5 to 8 (4x^3 - x) dx, n=6 The left Riemann sum, Ln, is (Do not round until the final answer. Then round to three decimal places as needed.)

Find Ln, Rn, and their average for the definite integral below using the indicated value of n.

∫ from 5 to 8 (4x^3 - x) dx, n=6

The left Riemann sum, Ln, is
(Do not round until the final answer. Then round to three decimal places as needed.)

Transcript text: Find $L_{n}, R_{n}$, and their average for the definite integral below using the indicated value of $n$. \[ \int_{5}^{8}\left(4 x^{3}-x\right) d x, n=6 \] The left Riemann sum, $L_{n}$, is $\square$ (Do not round until the final answer. Then round to three decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the width of each subinterval, $\Delta x$

The width of each subinterval is $\Delta x = \frac{b - a}{n} = \frac{8 - 5}{6} = 0.5.$

Step 2: Compute $L_{n}$ (Left Riemann Sum)

For each subinterval, calculate the left endpoint's value of $f(x)$ and multiply by $\Delta x$. Sum these areas to get $L_{n} = 3075$.

Step 3: Compute $R_{n}$ (Right Riemann Sum)

For each subinterval, calculate the right endpoint's value of $f(x)$ and multiply by $\Delta x$. Sum these areas to get $R_{n} = 3847.5$.

Step 4: Calculate the average of $L_{n}$ and $R_{n}$

The average of $L_{n}$ and $R_{n}$ is $\frac{L_{n} + R_{n}}2 = 3461.25$.

Final Answer:

The Left Riemann Sum approximation is $L_{n} = 3075$, the Right Riemann Sum approximation is $R_{n} = 3847.5$, and their average value is 3461.25.

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