 #1
 142
 26
Summary:

Given
##f(x) =
\begin{cases}
5 & \quad \text{if } x \text{ <3}\\
7 & \quad \text{if } x \geq3
\end{cases}##
with partitioning ##Pn=[0,3−\frac{1}{n},3+\frac{1}{n},4]## where n∈N and ##I=[0,4]##.
Is the function integrable on I?
So, I know that a function is integrable on an interval [a,b] if
##U(f,P_n)L(f,P_n)<\epsilon ##
So I find ##U(f,P_n## and ##L(f,P_n##
##L(f,P_n)=5(3\frac{1}{n}0)+5(3+\frac{1}{n}(3\frac{1}{n}))+7(4(3+\frac{1}{n}))=22\frac{2}{n} ##
##U(f,P_n)=5(3\frac{1}{n}0)+7(3+\frac{1}{n}(3\frac{1}{n}))+7(4(3+\frac{1}{n}))=22+\frac{2}{n}##
Then ##U(f,P_n)L(f,P_n)=\frac{4}{n} ##
So I'm struggeling to see if this fraction is less than epsilon or not. The "n" is confussing me. I have done similar tasks, but then I was given a fixed partition expressed with epsilon in the text. Some of my fellow students have tried to explain, but I still don't understand. Does someone have a good explanation?
##U(f,P_n)L(f,P_n)<\epsilon ##
So I find ##U(f,P_n## and ##L(f,P_n##
##L(f,P_n)=5(3\frac{1}{n}0)+5(3+\frac{1}{n}(3\frac{1}{n}))+7(4(3+\frac{1}{n}))=22\frac{2}{n} ##
##U(f,P_n)=5(3\frac{1}{n}0)+7(3+\frac{1}{n}(3\frac{1}{n}))+7(4(3+\frac{1}{n}))=22+\frac{2}{n}##
Then ##U(f,P_n)L(f,P_n)=\frac{4}{n} ##
So I'm struggeling to see if this fraction is less than epsilon or not. The "n" is confussing me. I have done similar tasks, but then I was given a fixed partition expressed with epsilon in the text. Some of my fellow students have tried to explain, but I still don't understand. Does someone have a good explanation?