Consider the function f(x) = x^{2} on the interval [1, 9]. Let P be a uniform partition of [1,9] with 16 sub-intervals. Compute the left and right Riemann sum of f on the partition. Use exact values.

Khadija Wells 2021-01-31 Answered
Consider the function f(x)=x2 on the interval [1, 9]. Let P be a uniform partition of [1,9] with 16 sub-intervals. Compute the left and right Riemann sum of f on the partition. Use exact values.
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Mayme
Answered 2021-02-01 Author has 103 answers

To find left and rigth Riemann sum of f(x)=x2 on the interval [1, 9] with 16 subintervals. The formula for left Riemann sum is, abf(x)dxΔx(f(x0)+f(x1)+....f(xn2)+f(xn1)
where Δx=ban
Here f(x)=x2 and [a,b]=[1,9],n=16
So Δx=9116=12.
Now divide the interval [1, 9] into n=16 subintervals of the length Δx=12.
The subintervals are,
[1,32],[32,2],[2,52],[52,3],[3,72],[72,4]....[8,172],[17/2,9].
To find left Riemann sum use the left end points of the subintervals.
Here the left endpoints are, 1, 32,2,....,8,172.
Now evaluate the function at left endpoints,
f(x0)=f(1)=12=1
f(x1)=f(32)=(32)2=92
f(x2)=f(2)=22=4
f(x3)=f(52)=(52)2=255

f(x14)=f(8)=82=64
f(x15)=f(172)=(172)2=2894
Substitute all values in the formula,
19x2dx12(1+94+4+254+9+494+....+64+2894)
12(1+2.25+4+6.25+9+....+64+72.25)
12(446)
223
Therefore left Riemann sum is 223.
To find right Riemann sum.
The formula for right Riemann sum is,
abf(x)dxΔx(f(x1)+f(x2)+....f(xn1)+f(xn)) where
Δx=ban
Here f(x)=x2and[a,b]=[1,9],n=16.
So Δx=9116=12.
Now divide the interval [1, 9] into n=16 subintervals of the length
Δx=.12
The subintervals are,

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