If f(x) = 4x2 − 5x, 0 ≤ x ≤ 3, evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints. R6 =

Answer 1

R6 = 4(3)^2 -5(3) = 36-15 = 11

R5 = 4(2.5)^2 -5(2.5) = 25-12.5 = 12.5

R4 = 4(2)^2 -5(2) =16-10 = 6

R3 = 4(1.5)^2 -5(1.5) = 9-7.5 =1.5

R2 = 4(1)^2 -5(1) =4-5 = -1

R1 = 4(.5)^2 -5(.5) = 1 -1.25 = -0.25

11+12.5 +6 + 1.5 = 31

-1 -.25 = -1.25

31-1.25= 29.75

29.75(3/6)= 14.875

equals area under the curve minus area above the curve when it's below the x axis, (31-1.25)/2=14.875

using integrals calculates 15 3/4 as the area under the curve by subtracting area under the x axis from the area above the x axis. It has to be calculated with two separate integrals evaluated from 0 to 5/4 and from 5/4 to 3, as the curve crosses the x axis at x=5/4

net area = 21- 21/4 = 15.75