Prove f is integrable using six subintervals and evaluate.

1. Apr 11, 2010

gotmilk04

1. The problem statement, all variables and given/known data
Define f as: f(x)= 2 if 0$$\leq$$x<1
f(1)=0
f(x)= -1 if 1<x<2
f(2)= 3
f(x)=0 if 2<x<3
f(3)=1
Prove f is integrable using six subintervals and find the value of $$\int$$f(x) dx

3. The attempt at a solution
Let P={0, 1-h, 1+h, 2-h, 2+h, 3-h, 3} where 0<h<1/2
I just need help proving $$\int$$f(x) dx=1.
I can prove I$$\geq$$1, but I'm having trouble proving J<1+$$\epsilon$$
I have S$$_{P}$$=2(1-h)+2(2h)+(-1)(1-2h)+3(2h)+0(1-2h)+h= 2+9h
but now I don't know how to get that to less than 1+$$\epsilon$$

2. Apr 11, 2010

Staff: Mentor

I would split the interval up this way:
P = {0, 1 - h, 1, 1 + h, 2 - h, 2, 3}

I'm basically ignoring the discontinuities at 2 and 3, since they aren't going to contribute anything to the integral.