Show that f is integrable

1. The problem statement, all variables and given/known data
Suppose that f(x)=c for all x in [a,b]. Show that f is integrable and that [tex]\int ^{a}_{b}[/tex]f(x)dx = c(b-a)


2. Relevant equations



3. The attempt at a solution
I understand all the definitions for Integration, my problem lies with approaching the problem. Should I use first principles to solve the problem? or do I just need to quote the definition?

For example, can I say "since f(x) is constant then it must be the case that upper sums equal its lower sums. Which implies that its upper integral equals its lower integral. Therefore f is Riemann integrable. Such that [tex]\int ^{a}_{b}[/tex]f(x)dx = c(b-a)

Is this sufficient?

Thank you for your help.

M

 

Thank you for your reply.

So my attempted solution is correct? All I need to do now is to prove that it is indeed c(b-a)?

So since c is a constant, it follows that c=M=m, where M and m are the max and min of each interval in the partion P[tex]\epsilon[/tex][a,b]. This therefore implies that we do not need any partitions in [a,b] since its a straight line. U(f,P) = L(f,P) therefore c(b-a).

Thanks,

M

 

All right, thank you for your help.