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 #?
1 Answer
# LRS = -6 #
# R RS = =18 #
Explanation:
We have:
# f(x) = 3-1/2x #
We want to calculate over the interval
# 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
