# Thread: Prove that a function is Riemann integrable?

1. ## Prove that a function is Riemann integrable?

Consider the function f(x)=-x+2, 0<=x<=1 on the interval [0,1]

1)Let Pn be the partition {0,1/n,2/n,...,1}.Calculate L(Pn) (the lower sum) and U(Pn) (the upper sum).
I know the formula for L(P) and U(P) but I don't really understand it and I'm having trouble applying it. I know it's the sum of m=(i-1)/n and M=(i/n) but I don't know how to adapt this to the formula.Can anyone help please?

2)Hence prove that f is Riemann Integrable on [0,1] and evaluate the integral?
Once I have U(P) and L(P) I find the limit as n tends to infinity and if the limits are equal the function is integrable?and the value of the integral is -1/2x^2+2x+c.

Can someone please correct me where I'm wrong and help me with the rest.

Thanks in advance.

2. $U(P) = \sum\limits_{k = 1}^n {f\left( {\frac{k}
{n}} \right)\Delta _x } = \sum\limits_{k = 1}^n {\left( {\frac{k}
{n} + 2} \right)\frac{1}
{n}} = \frac{1}
{n}\left( {\frac{{n\left( {n + 1} \right)}}
{2n} + 2n} \right) = \frac{{5n + 2}}
{{2n}}$

3. I understand why it would be like that for a positive function but for a negative straight line wouldn't that be the formula for L(P) and wouldn't it be -(k/n)+2? and the because -x+2 is a straight line wouldn't U(P) be the sum from 0 to (n-1)/n and how would you then derive a formula for the sum where the terms are negative?

4. Yes, you are correct. I simply did not seen that - sign.
That is a good reason to learn to post your questions in LaTex.
As you have written it, it is very hard to read.

5. I apologise for the misunderstanding but thank you so much for clarifying this for me