Question

In this example of integration with Riemann sums http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/part-a-definition-of-the-definite-integral-and-first-fundamental-theorem/session-46-riemann-sums/MIT18_01SCF10_ex46sol.pdf

what does c(i) stand for?

And what is the formula for c(i) derived from?

Answer 1

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Answer 2

Try again, removed the ( )

Answer 3

it defines c(i) in the first sentence of the solution part.

Answer 4

Yes what I'm having trouble is, is comprehending where \[c_i=\frac{i-1}{3}\] came from and what i t represents

Answer 5

the "i" is just a notation for iteration

notice that they arbitrarily decided to divvy up the interval into widths of 1/3

Answer 6

the left endpoint is denoted by ci, and we can measure this from the value of the left side of the interval

at "a" (the leftmost value of the interval); c1 = a; and f(c1) is the height

c2 = a+1/3
c3 = a+2/3
c4 = a+3/3

each time we increase the distance from "a" by an iteration of 1/3 agreed?

Answer 7

for this problem; the interval is from 0 to 2; so, a=0. giving us ci = i/3

for some reason, they are wanting to define the left most value of each interval as ci such that i starts at a value of 1 therefore\[c_i=\frac{i-1}{3}={\{0,\frac13,\frac23,3,\frac43,...\}}~:~i=1,2,3,...\]

Answer 8

hmm this is actually quite obvious come to think of it.

Correct me if I'm wrong, this means \[c_i\] is just the input variable for .. well \[f(c_i)\] which determines height.

Answer 9

correct

Answer 10

alright, wonder how I missed that.

Answer 11

Thank you for your time :)

Answer 12

youre welcome :)