1—Basic Stuff 13
a b
x 1 x 2
ξ 1 ξ 2 ξN
- PickN− 1 points betweenaandb. Call themx 1 ,x 2 , etc.
a=x 0 < x 1 < x 2 <···< xN− 1 < xN=b
where for convenience I label the endpoints asx 0 andxN. For the sketch ,N= 8.
- Let∆xk=xk−xk− 1. That is,
∆x 1 =x 1 −x 0 , ∆x 2 =x 2 −x 1 ,···
- In each of theN subintervals, pick one point at which the function will be evaluated. I’ll label these points
by the Greek letterξ. (That’s the Greek version of “x.”)
xk− 1 ≤ξk≤xk
x 0 ≤ξ 1 ≤x 1 , x 1 ≤ξ 2 ≤x 2 ,···
- Form the sum that is an approximation to the final answer.
f(ξ 1 )∆x 1 +f(ξ 2 )∆x 2 +f(ξ 3 )∆x 3 +···
- Finally, take the limit as all the∆xk → 0 and necessarily then, asN → ∞. These six steps form the
definition
lim
∆xk→ 0
∑N
k=1
f(ξk)∆xk=
∫b
a
f(x)dx
