Calculate the left Riemann sum for the given function over the given interval, using the given value of n. (When rounding, round answers to four decimal places.)
f(x) = e^−x over [−6, 6], n = 6
To calculate the left Riemann sum for the function f(x) = e^(-x) over the interval [-6, 6] using n = 6, we can follow these steps:
1. Determine the width of each subinterval. The interval [-6, 6] is divided into n = 6 subintervals, so the width of each subinterval is (b - a) / n, where a is the left endpoint (-6) and b is the right endpoint (6). In this case, the width is (6 - (-6)) / 6 = 12 / 6 = 2.
2. Identify the left endpoint of each subinterval. Starting from the left endpoint (-6), we move in increments of the width (2) to get the left endpoints of each subinterval. The left endpoints for n = 6 are: -6, -4, -2, 0, 2, 4.
3. Evaluate the function at each left endpoint. We plug in the left endpoints into the function f(x) = e^(-x) and calculate the value.
For the left Riemann sum, we sum up the function values multiplied by the width of each subinterval.
Let's calculate it step by step:
- For the first subinterval [-6, -4]:
- Left endpoint: -6
- Function value: f(-6) = e^(-(-6)) = e^6 (approximately 403.4288)
- For the second subinterval [-4, -2]:
- Left endpoint: -4
- Function value: f(-4) = e^(-(-4)) = e^4 (approximately 54.5982)
- For the third subinterval [-2, 0]:
- Left endpoint: -2
- Function value: f(-2) = e^(-(-2)) = e^2 (approximately 7.3891)
- For the fourth subinterval [0, 2]:
- Left endpoint: 0
- Function value: f(0) = e^(-0) = e^0 = 1
- For the fifth subinterval [2, 4]:
- Left endpoint: 2
- Function value: f(2) = e^(-2) (approximately 0.1353)
- For the sixth subinterval [4, 6]:
- Left endpoint: 4
- Function value: f(4) = e^(-4) (approximately 0.0183)
Now, we calculate the left Riemann sum by summing up these function values multiplied by the width of each subinterval:
(2 * e^6) + (2 * e^4) + (2 * e^2) + (2 * 1) + (2 * 0.1353) + (2 * 0.0183)
Simplifying further, we get:
2 * (e^6 + e^4 + e^2 + 1 + 0.1353 + 0.0183)
Now, you just need to calculate this expression to obtain the left Riemann sum.