Work

[AmySaunders]

The table shows values of a force function f(x), where x is measured in meters and f(x) in newtons. Use the Midpoint Rule with n = 4 to estimate the work W done by the force in moving an object from x = 5 to x = 21.

x	5	7	9	11	13	15	17	19	21
f(x)	5	5.7	7.3	8.6	9.6	8	6.5	5.2	3.9

I tried using the midpoint rule with 4 subintervals, and came out with 4(6.15 + 8.45 + 8.05 + 5.2)

which equals 111.40

which is not correct. Where am I going wrong?

 

[HallsofIvy]

The work done by variable force f(x) across distance x if $\displaystyle \int f(x) dx$. Since you are dividing 5 to 21 into 4 intervals, each of length (21- 5)/4= 4, the endpoints are 5, 9, 13, 17, and 21 and the midpoints are 7, 11, 15, and 19. f(7)= 5.7, not 6.15, f(11)= 8.6, not 8.45, f(15)= 8.0, not 8.05. You have f(19)= 5.2 but I don't know where you got those other numbers.

 

[AmySaunders]

I used my endpoints to get an exact midpoint between each set of endpoints.

For example:

f(9)=7.3 and f(5)=5

7.3-5=2.3

2.3/2=1.15

5+1.15=6.15 which is the exact middle between 5 and 9.

If I was to use those numbers given to me in the tables, what would I do then? I multiplied each position by it's corresponding coordinate and came out with 353.4 which is still wrong.

 

[Deleted member 4993]

The table shows values of a force function f(x), where x is measured in meters and f(x) in newtons. Use the Midpoint Rule with n = 4 to estimate the work W done by the force in moving an object from x = 5 to x = 21.

x	5	7	9	11	13	15	17	19	21
f(x)	5	5.7	7.3	8.6	9.6	8	6.5	5.2	3.9

I tried using the midpoint rule with 4 subintervals, and came out with 4(6.15 + 8.45 + 8.05 + 5.2)

which equals 111.40

which is not correct. Where am I going wrong?

For four sub-intervals your intervals should be x_o = 5, x₁ = 9, x₂ = 13, x₃ = 17 & x₅ = 21 with $\displaystyle \delta x$ = 4

Then the area under the curve is = 4 * (5.7 + 8.6 + 8 + 5.2) = 4 * 27.5 = 110

 

[AmySaunders]

For four sub-intervals your intervals should be x_o = 5, x₁ = 9, x₂ = 13, x₃ = 17 & x₅ = 21 with $\displaystyle \delta x$ = 4

Then the area under the curve is = 4 * (5.7 + 8.6 + 8 + 5.2) = 4 * 27.5 = 110

Thank you, thank you! I was being an idiot and I didn't see that I wasn't supposed to multiply each location by corresponding coordinate. When I multiply by 4- which is the number of subintervals- I get the right answer, just using the coordinates they gave me.

Thank you!