Advanced High-School Mathematics

250 CHAPTER 5 Series and Differential Equations

U(f;P) =

∑n

i=i

Mi(xi−xi− 1 ),

and the lower Riemann sum relative to the above partition is the
sum

L(f;P) =

∑n

i=i

mi(xi−xi− 1 ).

Before continuing, we need two more fundamental concepts, the
least upper boundandgreatest lower boundof a set of real num-
bers. Namely, ifA⊆R, we set

LUB(A) = mind {d≥a|a∈A},GLB(A) = maxd {d≤a|a∈A}.

Finally, we define the sets

U(f) ={U(f;P)|P is a partition of [a,b]},

L(f) ={L(f;P)|P is a partition of [a,b]}.

Definition. If LUB(L(f)) and GLB(U(f)) both exist, and if
LUB(L(f)) = GLB(U(f)), we say that f is Riemann integrable
over [a,b] and call the common value theRiemann integral of f
over the interval[a,b].

Example. Consider the function f(x) = x^3 , 0 ≤ x ≤ 2, and con-
sider the partition of [0,2] intonequally-spaced subintervals. Thus, if
P : 0 = x 0 < x 1 < ···xn = 2 is this partitiion, then xi =^2 ni, i =
0 , 1 , 2 ,...,n. Since f is increasing over this interval, we see that the
maximim off over each subinterval occurs at the right endpoint and
that the minimum occurs at the left endpoint. It follows, therefore,
that

U(f;P) =

∑n

i=1

( 2 i

n

) 3

·

2

n

=

16

n^4

∑n

i=1

i^3 , L(f;P) =

n∑− 1

i=0

( 2 i

n

) 3

·

2

n

=

16

n^4

n∑− 1

i=0

i^3.