Riemann Sum Calculation for f(x)=x on [0,2] with n=8

[Firben]

Let Pn denote the partition of the given interval [a,b] into n sub intervals of equal length Δxi = (b-a)/n
Evaluate L(f,Pn) and U(f,Pn) for the given functions f and the given values of n.

f(x)=x on [0,2], with n=8

2.My solution

x0 = 0, x1 = 1/4, x2 = 1/2, x3 = 3/4, x4 = 1, x5 = 5/4, x6 = 3/2, x7 = 7/4, x8 = 2

L(f,Pn) = 1/4(1/4+1/2+3/4+1+5/4+3/2+7/4+2) = 2.25 = 9/4

U(f,Pn) = 1/4(0+1/4+1/2+3/4+1+5/4+3/2+7/4) = 1.75 = 7/4

In the answersheet the lower rienmann sum is 7/4 and the upper rienmann is 9/4

What is wrong ?

 

[LCKurtz]

Let Pn denote the partition of the given interval [a,b] into n sub intervals of equal length Δxi = (b-a)/n
Evaluate L(f,Pn) and U(f,Pn) for the given functions f and the given values of n.

f(x)=x on [0,2], with n=8

2.My solution

x0 = 0, x1 = 1/4, x2 = 1/2, x3 = 3/4, x4 = 1, x5 = 5/4, x6 = 3/2, x7 = 7/4, x8 = 2

L(f,Pn) = 1/4(1/4+1/2+3/4+1+5/4+3/2+7/4+2) = 2.25 = 9/4

U(f,Pn) = 1/4(0+1/4+1/2+3/4+1+5/4+3/2+7/4) = 1.75 = 7/4

In the answersheet the lower rienmann sum is 7/4 and the upper rienmann is 9/4

What is wrong ?

The only thing I see that is incorrect is your spelling of "Riemann".

 

[micromass]

L(f,Pn) = 1/4(1/4+1/2+3/4+1+5/4+3/2+7/4+2)
U(f,Pn) = 1/4(0+1/4+1/2+3/4+1+5/4+3/2+7/4)

Shouldn't these two be swapped?? To find L, you take the lowest value in the interval. So it makes sense that

L(f,Pn) = 1/4(0+1/4+1/2+3/4+1+5/4+3/2+7/4)

and not the thing you wrote.

 

[LCKurtz]

Shouldn't these two be swapped??

Heh. I didn't even notice that and I was wondering why he said the answers were wrong.