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 #?
# LRS = 6 #
# R RS = =18 #
We have:
# f(x) = 31/2x #
We want to calculate over the interval
# Deltax = (142)/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 \ 31/2x \ dx #
# " " = [ 3xx^2/4 ]_2^14 #
# " " = (4249)  (61) #
# " " = (7)  (5) #
# " " = 12 #
Note that as the function is linear the exact valkuye is the average of the LRS and the RRS
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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:

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).

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]).

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)

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)
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