I found a problem in a paper which has asked below two questions.
QUESTION
Let $f : [0,1] -> \mathbb{R} $ be the fucntion given by $f(x) = x^2+x $ for all $x\in [0,1]$
1) Use the partition $P=[0,1/4,1/2,3/4,1]$ to find an Upper bound and a lower bound for the area of the region lying between the graph $f(x)$ and the $x-$ axis.
2) Without calculating lower and upper Riemann integrals, can you check the riemann integrability of $f(x)$ on $[0,1]$? Give reasons.
WHAT I FOUND
As question has asked I tried to find rimann left and right sums which will give the upper riemann integral and lower one. After the calculation has done I found that Upper rimann sum is $1.09375$ and lower as $0.59375$.
MY Problem
There is only one partition is given. So obviously Upper rimann sum is equals to the Upper riemann integral and lower rimann sum is equals to lower riemann integral. (If I'm wrong correct me).
But that says this function is not riemann integrable. Which makes me confused about my result. (I also tried to verify result by calculating sums using online calculator and got the same result)
And also Since this function is monotonically increasing in $[0,1]$ interval, definitely this should be riemann integrable (By Theory)
So please can someone suggest or explain me why is that or what I've done incorrectly? Thank you.
Best Answer
This is your problem:
You correctly found that the upper and lower sums for this one partition are not equal (assuming your arithmetic is right - I didn't check). But when you look at other partitions you will find different bounds. When you consider all the partitions you'll find that the smallest number larger than all the lower sums is the same as the largest number smaller than all the upper sums. That's the definition of Riemann integrability.