Calculate the Riemann Sum for the following equation

Question

The function is f(x) = 1/x and the interval I = [1, 4]. Partition the interval using the points

x₀ = 1, x₁ = 2, x₂ = 5/2 , x₃ = 7/2 , x₄ = 4.

Use the midpoint in each of the subintervals as a distinguished point.

Answer 1

construct 4 rectangles, calculate the area of each, and sum their areas

x0 to x1 has base = 1, height =1/(3/2)= 2/3, ,area = 2/3, midpoint is 1 1/2 = 3/2

x1 to x2 has base 1/2, height 1/(9/4)= 4/9, area = 2/9 midpoint is 2 1/4 = 9/4

x2 to x3 has base 1, height 1/3, area= 1/3, midpoint is 3

x3 to x4 has base 1/2, height 1/(15/4)= 4/15, midpoint is 3 3/4= 15/4

add them

2/3 + 2/9+ 1/3 + 4/15= 1+ (10+12)/45

= 1 22/45

as a rough check, calculate the definite integral of 1/x evaluated between 1 and 4

ln4 - ln1 = ln4 + 0 = ln4 = about 1.386 which is very close to 1 22/45 =about 1.489