Find this difference without actually calculating either of the Riemann sums.

Question

Consider the area of the region bounded above by the graph of f(x) = x²,

 bounded below by the x-axis, and between the lines x = 3

 and x = 5.

 If a regular partition with 100 subintervals is used, what is the difference between the value of the left Riemann sum and the value of the right Riemann sum? Find this difference without actually calculating either of the Riemann sums.

R100-L100

Answer 1

Answer 2

integral of x^2 = x^3/3

evaluated between x=3 and 5

= 5^3/3 - 3^3/3 = (125-27)/3 = 98/3 = 32 2/3

100 subdivisions make both left and right estimates very close to the actual 32 2/3

so their difference is also very small

the more subdivisions the smaller the differences. 100 should make the difference relatively small.

calculate the left most interval f(3) = 9 multiplied by (5-3)100 = 1/50= .09

height of the rectangle = f(3) = 3^2 =9, base of the rectangle = 1/50. area = 18/100 =.18

and the right most interval area f(5) = 5^2 = 25 x 1/50 = .5

all other rectangle areas, the other 99 are the same for left or right

so the difference in the left and right estimates is .5-.18 = .32

R100 - L100 = 0.32