How do you calculate the left and right Riemann sum for the given function over the interval [2,14], n=6 for # f(x)= 3 - (1/2)x #?

Question

Scarlett Allison · Accepted Answer

# LRS = -6 #
# R RS = =18 #

We have:

# f(x) = 3-1/2x #

We want to calculate over the interval #[2,14]# with #6# strips; thus:

# Deltax = (14-2)/5 = 2#

Note that we have a fixed interval (strictly speaking a Riemann sum can have a varying sized partition width). The values of the function are tabulated as follows;

Left Riemann Sum

# LRS = sum_(r=0)^5 f(x_i) \ Deltax_i #
# " " = 2 * (2 + 1 - 0 - 1 - 2 - 3) #
# " " = 2 * (-3) #
# " " = -6 #

Right Riemann Sum

# R RS = sum_(r=1)^6 f(x_i) \ Deltax_i #
# " " = 2 * (1 - 0 - 1 - 2 - 3 - 4) #
# " " = 2 * (-9) #
# " " = -18 #

Actual Value

For comparison of accuracy:

# Area = int_2^14 \ 3-1/2x \ dx #
# " " = [ 3x-x^2/4 ]_2^14 #
# " " = (42-49) - (6-1) #
# " " = (-7) - (5) #
# " " = -12 #

Note that as the function is linear the exact valkuye is the average of the LRS and the RRS

Charles Arocha · Answer

undefined To calculate the left and right Riemann sums for the function $f(x) = 3 - \frac{1}{2}x$ over the interval $[2,14]$ with $n=6$ subdivisions:

1. **Interval Length $\Delta x$**: 
   The total interval length is $14 - 2 = 12$. With $n = 6$ subdivisions, each subinterval has length $\Delta x = \frac{12}{6} = 2$.

2. **Determine Subinterval Endpoints**:
   The subintervals are defined by splitting the interval $[2,14]$ into 6 equal parts, each of length 2. These intervals are: $[2,4], [4,6], [6,8], [8,10], [10,12], [12,14]$.

3. **Left Riemann Sum (LRS)**:
   For the Left Riemann Sum, use the left endpoints of each subinterval to find the heights of the rectangles. The left endpoints are $x = 2, 4, 6, 8, 10, 12$.
   - At $x = 2$, $f(x) = 3 - \frac{1}{2}(2) = 2$
   - At $x = 4$, $f(x) = 3 - \frac{1}{2}(4) = 1$
   - At $x = 6$, $f(x) = 3 - \frac{1}{2}(6) = 0$
   - At $x = 8$, $f(x) = 3 - \frac{1}{2}(8) = -1$
   - At $x = 10$, $f(x) = 3 - \frac{1}{2}(10) = -2$
   - At $x = 12$, $f(x) = 3 - \frac{1}{2}(12) = -3$
   
   LRS = $\Delta x \times \text{sum of the function values at these points}$
   
   LRS = $2 \times (2 + 1 + 0 - 1 - 2 - 3) = 2 \times (-3) = -6$

4. **Right Riemann Sum (RRS)**:
   For the Right Riemann Sum, use the right endpoints of each subinterval to find the heights of the rectangles. The right endpoints are $x = 4, 6, 8, 10, 12, 14$.
   - At $x = 4$, $f(x) = 1$
   - At $x = 6$, $f(x) = 0$
   - At $x = 8$, $f(x) = -1$
   - At $x = 10$, $f(x) = -2$
   - At $x = 12$, $f(x) = -3$
   - At $x = 14$, $f(x) = 3 - \frac{1}{2}(14) = -4$
   
   RRS = $\Delta x \times \text{sum of the function values at these points}$
   
   RRS = $2 \times (1 + 0 - 1 - 2 - 3 - 4) = 2 \times (-9) = -18$