Consider the function f(x) = sin x on th interval [0, 3]. Let P be a uniform partition of [0, 3] with 4 sub-intervals. Compute the left and right Riemann sum of f on the partition. Enter approximate values, rounded to three decimal places.

Braxton Pugh 2021-02-26 Answered
Consider the function f(x)=sinx on th interval [0, 3]. Let P be a uniform partition of [0, 3] with 4 sub-intervals. Compute the left and right Riemann sum of f on the partition. Enter approximate values, rounded to three decimal places.
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dieseisB
Answered 2021-02-27 Author has 85 answers

Let f(x) be defined on the closed interval [a,b] and let [a,b] is portioned into n subintervals. Then, the left Riemann sum is defined as i=0n1f(xi)Δx
where xi is the left endpoint of each subinterval and the right Reimann sum is
i=0nf(xi)Δx where xi is the right endpoint of each subinterval.
The function is (x)=sinx on the interval [0, 3]. Here the interval is portioned into 4 subintervals. That is, n=4.
Find the length of each subinterval Δ x as,
Δx=ban
=304
=34
=0.75
Therefore the subintervals are, [0,0.75],[0.75,1.5],[1.5,2.25],[2.25,3].
Left sum:
Find the left Reimann sum as follows.
i=0n1f(xi)Δx=0.75(f(x+0)+f(x1)+f(x2)+f(x3))
=0.75(sin(0)+sin(0.75)+sin(1.5)+sin(2.25))
1.843
Right sum:
Find the Right Reimann sum as follows.
i=1nf(xi)Δx=0.75(f(x1)+f(x2)+f(x3)+f(x4))
=0.75(sin(0.75)+sin(1.5)+sin(2.25)+sin(3))
1.949
Left sum: 1.843
Right sum: 1.949

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