The rectangles in the graph below illustrate a left and then

The rectangles in the graph below illustrate a left and then a right endpoint Riemann sum for f(x)=(-(x^2)/6)+2x on the interval (4,8)

Using the left and right Riemann sums above, we definitively conclude that +2r dr

Solution

Given f(x)=(-x^2/6)+2x on the interval (4,8)

To find left and right endpoints riemann sum we must substitute the given interval in the given f(x)

taking interval from x=4, f(x) = (-4^2)/6 + 2(4) = 32/3 = 10.66

x=4.5 f(x) = (-4.5^2)/6 + 2(4.5) = 12.375

x=5 f(x) =(-5^2)/6 + 2(5) = 14.16

x=5.5 f(x) = (-5.5^2)/6 + 2(5.5) = 16.04

x= 6 f(x) = (-6^2)/6 + 2(6) = 18

x=6.5 f(x) = (-6.5^2)/6 + 2(6.5) = 20.04

x= 7 f(x) = (-7^2)/6 + 2(7) = 22.17

x=7.5 f(x) =(-7.5^2)/6 + 2(7.5) =23.375

x=8 f(x) =(-8^2)/6 + 2(8) =26.67

1.. the first question is to find integral from 4 to 6

taking the above f(x) from 4 to 6

the left endpoints are given by 4,4.5 and 5 then f(x) are given by 10.66,12.375,14.16

taking these values sum and dividing it by 2 to get its area.

their area is given by 37.195/2 = 18.59 this is the left endpoint

the right endpoints are given by 5,5.5,6 then f(x) are given as 14.16,16.04,18

taking these values sum and dividing it by 2 to get its area.

their area is given by 48.2/2 = 24.1 this is the right endpoint

2.. the second question is to find integral from 6 to 8

taking the above f(x) from 6 to 8

the left endpoints are given by 6,6.5 and 7 then f(x) are given by 18,20.04,22.17

taking these values sum and dividing it by 2 to get its area.

their area is given by 30.105 this is the left endpoint

the right endpoints are given by 7,7.5,8 then f(x) are given as 22.17,23.375,26.67

taking these values sum and dividing it by 2 to get its area.

their area is given by 36.10 this is the right endpoint

3.. the first question is to find integral from 4 to 8

taking the above f(x) from 4 to 8

the left endpoints are given by 4,4.5,5,5.5 and 6 then f(x) are given by 10.66,12.375,14.16,16.04,18

taking these values sum and dividing it by 2 to get its area.

their area is given by 35.61 this is the left endpoint

the right endpoints are given by 6,6.5,7,7.5,8 then f(x) are given as 18,20.04,22.17,23.375,26.67

taking these values sum and dividing it by 2 to get its area.

their area is given by 55.12 this is the right endpoint.

From the above three left and right endpoints, left endpoint are less than right endpoints.