Rieman sums are used to calculate the approximate value of the area under the curve of real functions.
A left Riemann sum uses rectangles whose top-left vertices are on the curve while the right Riemann sum uses rectangles whose top-right vertices are on the curve.
The width of each rectangle is given by:
[tex]w=\frac{b-a}{n}[/tex]
Where a and b are the endpoints of the interval and n is the number of rectangles.
We are given the function:
y = ln x on the interval [2, 9]
And it's required to use n = 3 rectangles, thus:
[tex]w=\frac{9-2}{3}=2.333[/tex]
For the left Riemann sum, we need to calculate the values of y for:
x = 2
x = 2 + w = 4.333
x = 2 + 2w = 6.667
f(2) = ln 2 = 0.6931
f(4.333) = 1.4662
f(6.667) = 1.8972
The area of the rectangles is the height by the base:
A = 0.6931 x 2.333 + 1.4662 x 2.333 + 1.8972 x 2.333
A = 9.4640
For the right Rieman sum, we need to calculate the values of y for:
x = 2 + w = 4.333
x = 2 + 2w = 6.667
x = 2 + 3w = 9
f(4.333) = 1.4662
f(6.667) = 1.8972
f(9)= 2.1972
Calculate the area of the rectangles:
A = 1.4662 x 2.333 + 1.8972 x 2.333 + 2.1972 x 2.333
A = 12.973
Note: The real value for the area (using integration) is 11.39
