Is a Constant Function Always Riemann Integrable?

[michonamona]

Homework Statement

Suppose that f(x)=c for all x in [a,b]. Show that f is integrable and that \int ^{a}_{b}f(x)dx = c(b-a)

Homework Equations

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 \int ^{a}_{b}f(x)dx = c(b-a)

Is this sufficient?

Thank you for your help.

M

 

[Gib Z]

You should quote the definition from Riemann Integrability, and then explicitely substitute the information from your question into the definition. Your statement, even if it was sufficient to your professor to show f is intebrable, does nothing to show that the actual value of the integral is c(b-a).

 

[michonamona]

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\epsilon[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

 

[Gib Z]

No it is not. If you follow what I said in the last post, you will answer both parts of the question.

 

[michonamona]

All right, thank you for your help.