Calculate the left Riemann sums for the function over the given interval, using the given values of n. (When rounding, round answers to four decimal places.) $f ( x ) = e ^ { - x }$ over $[ - 5,5 ] , n = 5$
A)Left sum = 343.2693
B) Left sum = 171.6346
C) Left sum = 1,716.3464
D) Left sum = 46.4564
E) Left sum = 34.3269

Question

Calculate the left Riemann sums for the function over the given interval, using the given values of n. (When rounding, round answers to four decimal places.)   $f ( x ) = e ^ { - x }$ over $[ - 5,5 ] , n = 5$ 
A)Left sum = 343.2693
B) Left sum = 171.6346
C) Left sum = 1,716.3464
D) Left sum = 46.4564
E) Left sum = 34.3269

Quizplus · Accepted Answer

The left Riemann sum is given by the formula:$$L_n = \sum_{i=1}^n f(x_{i-1}) \Delta x$$where $f(x)$ is the function, $x_{i-1}$ is the left endpoint of the $i$th subinterval, and $\Delta x$ is the width of each subinterval.In this case, we have $f(x) = e^{-x}$, $[a,b] = [-5,5]$, and $n = 5$. So, the width of each subinterval is $\Delta x = \frac{b-a}{n} = \frac{5-(-5)}{5} = 2$.The left endpoints of the subintervals are:$$x_0 = -5, \quad x_1 = -3, \quad x_2 = -1, \quad x_3 = 1, \quad x_4 = 3$$So, the left Riemann sum is:$$L_5 = f(-5) \Delta x + f(-3) \Delta x + f(-1) \Delta x + f(1) \Delta x + f(3) \Delta x$$$$= e^{-(-5)} (2) + e^{-(-3)} (2) + e^{-(-1)} (2) + e^{(1)} (2) + e^{(3)} (2)$$$$= 343.2693$$Therefore, the left Riemann sum is 343.2693.

Calculate the Left Riemann Sums for the Function Over the Given