Prove f is integrable using six subintervals and evaluate.

[gotmilk04]

Homework Statement

Define f as: f(x)= 2 if 0\leqx<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 \intf(x) dx

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 \intf(x) dx=1.
I can prove I\geq1, 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

 

[Mark44]

Homework Statement

Define f as: f(x)= 2 if 0\leqx<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 \intf(x) dx

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 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.

I just need help proving \intf(x) dx=1.
I can prove I\geq1, 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