Questions: Consider the function f(x)=sin x on the interval [0,2]. Let P be a uniform partition of [0,2] with 6 sub-intervals. Compute the left and right Riemann sum of f on the partition. Enter approximate values, rounded to three decimal places. Left-sum: Right-sum:

Solution Steps

To compute the left and right Riemann sums for the function \( f(x) = \sin x \) on the interval \([0, 2]\) with a uniform partition of 6 sub-intervals, follow these steps:

1. Determine the width of each sub-interval, \(\Delta x\), by dividing the total interval length by the number of sub-intervals.
2. For the left Riemann sum, evaluate the function at the left endpoint of each sub-interval and multiply by \(\Delta x\). Sum these values.
3. For the right Riemann sum, evaluate the function at the right endpoint of each sub-interval and multiply by \(\Delta x\). Sum these values.

The interval \([0, 2]\) is divided into \(n = 6\) sub-intervals. The width of each sub-interval is calculated as: \[ \Delta x = \frac{b - a}{n} = \frac{2 - 0}{6} = \frac{1}{3} \approx 0.3333 \]

The left Riemann sum is computed by evaluating the function \(f(x) = \sin x\) at the left endpoints of each sub-interval: \[ \text{Left Sum} = \sum_{i=0}^{n-1} f(a + i \Delta x) \Delta x \] Calculating this gives: \[ \text{Left Sum} \approx 1.2514604479296212 \quad \text{(rounded to three decimal places: } 1.251\text{)} \]

The right Riemann sum is computed by evaluating the function \(f(x) = \sin x\) at the right endpoints of each sub-interval: \[ \text{Right Sum} = \sum_{i=1}^{n} f(a + i \Delta x) \Delta x \] Calculating this gives: \[ \text{Right Sum} \approx 1.5545595902048484 \quad \text{(rounded to three decimal places: } 1.555\text{)} \]

Final Answer