Suppose that the n-th Riemann sum (with n sub-intervals of equal length), for some function f(x) on the intrval [0, 2], is S_{n} = frac{2}{n^{2}} times sum_{k = 1}^{n} k. What is the value of int_{0}^{2} f(x) dx?

Daniaal Sanchez 2021-01-24 Answered
Suppose that the n-th Riemann sum (with n sub-intervals of equal length), for some function f(x) on the intrval [0, 2], is Sn=2n2 × k=1n k.
What is the value of 02 f(x) dx?
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saiyansruleA
Answered 2021-01-25 Author has 110 answers

Step 1
Sn=2n2 k=1n k
Sn=2n2 [1 + 2 + 3 +  + n]
Sn=2n2n(n + 1)2
Sn=n + 1n
f(x)=x + 1x
Step 2
Now to find
=02 f (x) dx
=02[xx + 1x]dx
=02(1 + 1x) dx
=[x + ln x]02
=(2 + ln 2)  (0 + ln 0)
=2 + ln 2

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