Thread: 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.

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?

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.

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