Question

Show that lim n->inifinity for the rieman sum of the following piecewise function exist (is 17/6)
f(x)= x^2 when 0 ≤ x≤ 1
=x+1 when 1 ≤ x ≤ 2

Obviously the answer to the integral of this question using rules is 17/6 but our teacher have asked us to show it using riemman sums.

Answer 1

for an integral \[\int\limits\limits_{a}^{b}f(x)dx\]
the Riemann sum is
\[\lim_{n \rightarrow \infty }\sum_{i=1}^{n}f(x _{i}^{*})\Delta x\]where \[\Delta x = \frac{ b-a }{ n }\] and \[x _{i}^{*} = a+(\Delta x)i\]
in your question you have piece wise function thus consider the first piece
\[f(x) = x ^{2} when 0\le x \le 1\]
here a = 0 and b = 1
\[\Delta x = \frac{ 1-0 }{ n } = \frac{ 1 }{ n }\]
\[x _{i}^{*} = 0+(\frac{ 1 }{ n })i = \frac{ i }{ n }\]
putting these values into Riemann sum formula
\[\lim_{n \rightarrow \infty }\sum_{i=1}^{n}f(\frac{ i }{ n })\frac{ 1 }{ n }\]
as f(x) =x^2 thus f(i/n) = (i/n)^2 therefore
\[\lim_{n \rightarrow \infty }\sum_{i=1}^{n}f(\frac{ i }{ n })\frac{ 1 }{ n } = \lim_{n \rightarrow \infty }\sum_{i=1}^{n}(\frac{ i }{ n })^{2}\frac{ 1 }{ n } = \lim_{n \rightarrow \infty }\sum_{i=1}^{n}(\frac{ i ^{2} }{ n ^{2} })\frac{ 1 }{ n } \]
\[ \lim_{n \rightarrow \infty }\sum_{i=1}^{n}(\frac{ i ^{2} }{ n ^{2} })\frac{ 1 }{ n } = \frac{ 1 }{ n } \lim_{n \rightarrow \infty }\sum_{i=1}^{n}(\frac{ i ^{2} }{ n ^{2} })\]

Answer 2

\[= \frac{ 1 }{ n ^{3} } \lim_{n \rightarrow \infty } \sum_{i=1}^{n}i ^{2}\]
as we know that \[\sum_{i=1}^{n}i ^{2 }= \frac{ n(n+1)(2n+1) }{6}\]
therefore
\[= \frac{ 1 }{ n ^{3} } \lim_{n \rightarrow \infty } \sum_{i=1}^{n}i ^{2}\]
\[=\lim_{n \rightarrow \infty }\frac{ 1 }{ n^3 }\times \frac{ n(n+1)(2n+1) }{ 6 }\]
\[= \lim_{n \rightarrow \infty } \frac{ 2n ^{3}+3n ^{2} +1}{ 6n ^{3} }\]
we know that when we take the infinite limit of rational functions with numerator and denominator having same degree thus the limit is evaluated by considering the coefficient of term having highest power.
thus we get
2/6 which are the coefficients of n^3
now 2/6 = 1/3---> answer for one piece of the given function.

Answer 3

now for the second piece
f(x) =x+1 for \[1 \le x \le 2\]
here a=1 and b=2
use the same formula as mentioned above to find \[\Delta x\] and \[x _{i }^{*}\]
i give you some useful formulas you will need it in your work
\[\sum_{i=1}^{n} i = \frac{ n(n+1) }{ 2 }\]
and

Answer 4

\[\sum_{i=1}^{n} i ^{2} = \frac{ n(n+1)(2n+1) }{ 6 }\]

Answer 5

when you will find this area add it with the previous area to get the entire area under the curve