How do you calculate the left and right Riemann sum for the given function over the interval [1,5], using n=4 for # f(x)= 3x#?
1 Answer
# LRS = 30 #
# R RS = 42 #
Explanation:
We have:
# f(x) = 3x #
We want to calculate over the interval
# Deltax = (5-1)/4 = 1#
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=1)^4 f(x)Deltax #
# " " = Deltax { f(1) + f(2) + f(3) + f(4) } \ \ \ # (The LHS values)
# " " = 1*(3+6+9+12) #
# " " = 30 #
Right Riemann Sum
# R RS = sum_(r=2)^5 f(x)Deltax #
# " " = Deltax { f(2) + f(3) + f(4) +f(5) } \ \ \ # (The RHS values)
# " " = 1*(6+9+12+15) #
# " " = 42 #
Actual Value
For comparison of accuracy:
# Area = int_1^5 \ 3x \dx #
# " " = 3[x^2/2]_1^5 #
# " " = 3/2{(25)-(1)} #
# " " = 36 #
