SOME ERROR ESTIMATES IN THE TRAPEZOIDAL QUADRATURE RULE

S. S. DRAGOMIR AND S. MABIZELA

Abstract. We derive sharp error bounds in the Trapezoidal Rule for functions
whose first derivative satisfies a strong continuity condition or whose second
derivative is an*Lp*function.

1. Introduction

The classical error estimate in the Trapezoidal Rule asserts that if a function*f*
has a bounded second derivative on the interval [*a, b*]*,* then the Trapezoidal Rule
converges to the true value of the definite integral R* _{a}bf*(

*x*)

*dx*with the rapidity of at least

*12*

_{n}*.*If the second derivative of

*f*is

*not*bounded, then the Trapezoidal Rule may or may not converge, and if it converges, it may do so very slowly.

In some practical problems the integrand does not have a bounded second de-rivative, and therefore the classical error estimate of the Trapezoidal Rule is not available. In this paper we derive some error bounds in the Trapezoidal Quadrature Rule for the following classes of functions:

(1) Functions whose first derivative belongs to the H¨older Space*Cα*_{[}_{a, b}_{]}_{,}_{where}

for*α _{∈}*(0

*,*1]

*,*

*Cα*[*a, b*] :=* _{{}f* : [

*a, b*]

_{→}_{R}:

*(*

_{|}f*x*)

*(*

_{−}f*y*)

*[*

_{| ≤}K_{|}x_{−}y_{|}α, x, y_{∈}*a, b*]

_{}}for some finite *K >*0*,*

(2) Functions whose second derivative is an*Lp−*function.

This paper is organized as follows: In this section we prove the key lemma (Lemma 1). This lemma is then applied to the classes of functions mentioned above.

In Section 2 we apply the results of Section 1 in deriving some inequalities of Trapezoid type.

In Section 3 we apply the results obtained in Section 2 to derive error bounds for the Trapezoidal Quadrature Rule for the above classes of functions.

Unless otherwise indicated, we shall assume throughout that [*a, b*] is a finite
interval.

The following lema will be useful in the sequel.

*Date*: May, 1999.

1991*Mathematics Subject Classification.* Primary 26D15; Secondary 41A55.

Lemma 1. *Let* *f* : [*a, b*]_{→}_{R} *be an absolutely continuous function on*[*a, b*]*.Then*
*we have the identity*

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*

= 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

[*f0*(*x*)* _{−}f0*(

*y*)] (

*x*)

_{−}y*dxdy.*

*Proof.* We have successively
Z *b*

*a*

Z *b*

*a*

[*f0*(*x*)* _{−}f0*(

*y*)] (

*x*)

_{−}y*dxdy*

=
Z *b*

*a*

Z *b*

*a*

[*xf0*(*x*) +*yf0*(*y*)* _{−}xf0*(

*y*)

*(*

_{−}yf0*x*)]

*dxdy*

= 2
Z *b*

*a*

Z *b*

*a*

[*xf0*(*x*)* _{−}xf0*(

*y*)]

*dxdy*

= 2
Z *b*

*a*

Z *b*

*a*

*xf0*(*x*)*dxdy _{−}*2
Z

*b*

*a*

Z *b*

*a*

*xf0*(*y*)*dxdy*

= 2 (*b _{−}a*)
"

*bf*(*b*)* _{−}af*(

*a*)

*Z*

_{−}*b*

*a*

*f*(*x*)*dx*
#

*−* *b*2* _{−}a*2[

*f*(

*b*)

*(*

_{−}f*a*)]

= (*b _{−}a*)2[

*f*(

*a*) +

*f*(

*b*)]

*2 (*

_{−}*b*) Z

_{−}a*b*

*a*

*f*(*x*)*dx.*

Dividing both sides by 2 (*b _{−}a*)2 yields the required result.

2. Some Integral Inequalities

Let us start to the following result for convex mappings.

Theorem 1. *Let* *f* : [*a, b*]_{→}_{R}*be a convex function on*[*a, b*]*.* *Then*

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx _{≥}*max

*(2.1)*

_{{|}A_{|},_{|}B_{|},_{|}C_{|}},*where*

*A* : = 1

*b _{−}a*
Z

*b*

*a*

*|xf0*(*x*)* _{|}dx_{−}* 1
(

*b*)2

_{−}aZ *b*

*a*

*|f0*(*x*)* _{|}dx_{·}*
Z

*b*

*a*

*|x _{|}dx,*

*B* : = 1

*b _{−}a*
Z

*b*

*a*

*x _{|}f0*(

*x*)

*+*

_{|}dx_{−}a*b*2

*·*

1
*b _{−}a*

Z *b*

*a*

*|f0*(*x*)_{|}dx,*and*

*C* : = 1

*b _{−}a*
Z

*b*

*a* *|*

*x _{|}f0*(

*x*)

*dx*(

_{−}f*b*)

*−f*(

*a*) (

*b*)2

_{−}aZ *b*

*a* *|*

*x _{|}dx.*

*Moreover, the inequality*(2*.*1) *is sharp in the sense that it cannot be replaced by a*
*smaller constant.*

*Proof.* Since*f* is a convex function on [*a, b*]*,*it follows that

for almost every*x, y _{∈}*(

*a, b*)

*.*Hence, for a.e.

*x, y*(

_{∈}*a, b*)

*,*

[*f0*(*x*)* _{−}f0*(

*y*)] (

*x*)

_{−}y= * _{|}*[

*f0*(

*x*)

*(*

_{−}f0*y*)] (

*x*)

_{−}y

_{|}*≥* max* _{{|}*[

*(*

_{|}f0*x*)

*(*

_{| − |}f0*y*)

*] [*

_{|}*]*

_{|}x_{| − |}y_{|}*[*

_{|},_{|}*(*

_{|}f0*x*)

*(*

_{| − |}f0*y*)

*] [*

_{|}*x*]

_{−}y

_{|},*|*[*f0*(*x*)* _{−}f0*(

*y*)] [

*]*

_{|}x_{| − |}y_{|}*Thus,*

_{|}}.Z *b*

*a*

Z *b*

*a*

[*f0*(*x*)* _{−}f0*(

*y*)] [

*x*]

_{−}y*dxdy*

*≥* max

(_{}

Z *b*

*a*

Z *b*

*a*

[* _{|}f0*(

*x*)

*(*

_{| − |}f0*y*)

*] [*

_{|}*]*

_{|}x_{| − |}y_{|}*dxdy*

*,*

Z *b*

*a*

Z *b*

*a*

[* _{|}f0*(

*x*)

*(*

_{| − |}f0*y*)

*] [*

_{|}*x*]

_{−}y*dxdy*

*,*

Z *b*

*a*

Z *b*

*a*

[*f0*(*x*)* _{−}f0*(

*y*)] [

*]*

_{|}x_{| − |}y_{|}*dxdy*)

*.*

A simple calculation shows that

*A* = 2

(*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

[* _{|}f0*(

*x*)

*(*

_{| − |}f0*y*)

*] [*

_{|}*]*

_{|}x_{| − |}y_{|}*dxdy,*

*B* = 2

(*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

[* _{|}f0*(

*x*)

*(*

_{| − |}f0*y*)

*] [*

_{|}*x*]

_{−}y*dxdy,*and

*C* = 2

(*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

[*f0*(*x*)* _{−}f0*(

*y*)] [

*]*

_{|}x_{| − |}y_{|}*dxdy.*

To prove the sharpness of the inequality (2*.*1)*,*let*f*(*x*) = 1* _{x}.*Then

*f*is differentiable and convex on (

*a, b*)

*,*where

*a >*0

*.*Now,

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*=*a*+*b*
2*ab* *−*

ln*b _{−}*ln

*a*

*b*

_{−}a
*,*

and

*|B _{|}* =

1
*b _{−}a*

Z *b*

*a*

1
*xdx−*

*a*+*b*
2 *·*

1
*b _{−}a*

Z *b*

*a*

1
*x*2*dx*

=

ln*b _{−}*ln

*a*

*b*

_{−}a*−*

*a*+*b*
2*ab*

=

*a*+*b*
2*ab* *−*

_{ln}_{b}

*−*ln*a*
*b _{−}a*

*.*

We therefore get equality in the inequality (2*.*1)*.*

The following inequality also holds.

Theorem 2. *Let* *f* : [*a, b*] _{→}_{R} *be an absolutely continuous function on* [*a, b*]*.* *If*

*f0 _{∈}Cα*[

*a, b*]

*,then we have the inequality*

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*
*≤*

*K*

(*α*+ 2) (*α*+ 3)(*b−a*)

*α+1*

*,*

*Proof.* Since*f0 _{∈}_{C}α*

_{[}

_{a, b}_{]}

_{,}_{there is a finite constant}

_{K >}_{0 such that}

*|f0*(*x*)* _{−}f0*(

*y*)

*for all*

_{| ≤}K_{|}x_{−}y_{|}α+1*x, y*[

_{∈}*a, b*]

*.*

Using Lemma 1, we have

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*

*≤* 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

*|f0*(*x*)* _{−}f0*(

*y*)

_{| |}x_{−}y_{|}dxdy*≤* *K*

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

*|x _{−}y_{|}α+1dxdy*

= *K*

(*α*+ 2) (*α*+ 3)(*b−a*)

*α+1*

*,*

where the last equality follows from the fact that
Z *b*

*a*

Z *b*

*a*

*|x _{−}y_{|}α+1dxdy* =
Z

*b*

*a*

Z *b*

*y*

(*x _{−}y*)

*α+1dx*+ Z

*y*

*a*

(*y _{−}x*)

*α+1dx*!

*dy*

= 1

*α*+ 2
Z *b*

*a*

h

(*b _{−}y*)

*α+2*+ (

*y*)

_{−}a*α+2*i

*dy*

= 2 (*b−a*)

*α+3*

(*α*+ 2) (*α*+ 3)*.*

and the theorem is proved.

The following corollary is natural.

Corollary 1. *Let* *f* : [*a, b*]_{→}_{R} *be an absolutely continuous function on* [*a, b*]*.* *If*

*f0* _{is Lipschitz continuous on}_{[}_{a, b}_{]}_{,}_{i.e.,}

*|f0*(*x*)* _{−}f0*(

*y*)

_{| ≤}K_{|}x_{−}y_{|}*for allx, y*[

_{∈}*a, b*]

*,*

*whereK >*0*,* *then we have the inequality*

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*
*≤*

*K*
12(*b−a*)

2

*.*

*Moreover, the constant* _{12}1 *is sharp.*

*Proof.* Take*α*= 1 in Theorem 2.

To prove the sharpness of the constant _{12}1*,*let*f*(*x*) =*x*_{2}2*.*Then

*|f0*(*x*)* _{−}f0*(

*y*)

*=*

_{|}

_{|}x_{−}y_{| ≤}K_{|}x_{−}y_{|}for all*x _{∈}*[

*a, b*]

*,*where

*K*= 1

*.*Consequently,

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*= 1

12(*b−a*)

2

*.*

and the proof is completed.

Theorem 3. *Let* *f* : [*a, b*]_{→}_{R} *be such that* *f0* _{is absolutely continuous on}_{[}_{a, b}_{]}_{.}

*If* *f00 _{∈}_{L}_{p}*

_{(}

_{a, b}_{)}

_{,}

_{where}_{1}

_{≤}_{p <}_{∞}_{,}_{then we have the estimation}

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*

*≤* 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

*Kp*(*x, y*)*|x−y|*
2*−*1

*p _{dxdy}*

*≤* *p*

2_{(}_{b}

*−a*)2*−p*1

(3*p _{−}*1) (4

*p*1)

_{−}*kf*

*00 _{k}*

*p,*

*where*

*Kp*(*x, y*) =

Z *y*

*x*

*|f00*(*t*)* _{|}pdt*
1

*p*

*, x, y _{∈}*[

*a, b*]

*.*

*The first inequality is sharp.*

*Proof.* By Lemma 1,

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*

= 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

[*f0*(*x*)* _{−}f0*(

*y*)] (

*x*)

_{−}y*dxdy*

= 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

Z *y*

*x*

*f00*(*η*)*dη*

(*x _{−}y*)

*dxdy.*

Also, using H¨older’s Inequality for *p >* 1*,*1*/p*+ 1*/q* = 1, we have that, for all
*x, y _{∈}*[

*a, b*]

*,*

Z *y*

*x*

*|f00*(*η*)_{|}dη*≤*

Z *y*

*x*

*|f00*(*η*)* _{|}pdη*
1

*p*

Z *y*

*x*

*dη*
1

*q*

=* _{|}x_{−}y_{|}*1

*q*

_{K}Now,

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*

*≤* 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

Z *y*

*x*

*f00*(*η*)*dη*

*|y−x|dxdy*

*≤* 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a* *|*

*x _{−}y_{|}*1

*−*1

*q*

Z *y*

*x* *|*

*f00*(*η*)* _{|}pdη*
1

*p*

*dxdy*

= 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a* *|*

*x _{−}y_{|}*2

*−*1

*p*

Z *y*

*x* *|*

*f00*(*t*)* _{|}pdt*
1

*p*

*dxdy*

*≤* 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

*|x _{−}y_{|}*2

*−*1

*p*

Z *b*

*a*

*|f00*(*t*)* _{|}pdt*
!1

*p*

*dxdy*

= *kf*

*00 _{k}*

*p*

2 (*b _{−}a*)2

*·*2

*p*2

_{(}

_{b}*−a*)4*−*1*p*

(3*p _{−}*1) (4

*p*1)

_{−}= *p*

2_{(}_{b}

*−a*)2*−*1*p*

(3*p _{−}*1) (4

*p*1)

_{−}*kf*

*00 _{k}*

*p.*

To prove the sharpness of the first inequality, let*f*(*x*) =*x*_{2}2*,*for*x _{∈}*[

*a, b*]

*.*Then

*f*(*a*) +*f*(*b*)

2 *−*

1
*b _{−}a*

Z *b*

*a*

*f*(*x*)*dx*
=

(*b _{−}a*)2
12

*,*

and

1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a*

*Kp*(*x, y*)*|x−y|dxdy*

= 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a* *|*

*x _{−}y_{|}p*1

*2*

_{|}_{x}_{−}_{y}_{|}*−*

1

*p _{dxdy}*

= 1

2 (*b _{−}a*)2
Z

*b*

*a*

Z *b*

*a* *|*

*x _{−}y_{|}*2

*dxdy*

= (*b−a*)

2

12 *.*

The case*p*= 1 goes likewise and we omit the details.

3. Error Bounds in the Trapezoidal Quadrature Rule

In this section we derive error bounds in the Trapezoidal Quadrature Rule for the classes of functions mentioned in Section 1.

Theorem 4. *Let* *f* : [*a, b*] _{→}_{R} *be an absolutely continuous function such that*

*f0* _{∈}_{C}α_{[}_{a, b}_{]}_{.}_{If}_{P}_{:}_{a}_{=}_{x}

0*< x*1*< x*2 *< ... < xn* =*b* *is a partition of the interval*

[*a, b*]*, hi*=*xi−xi−*1 *fori*= 1*,*2*, ..., n,and*

*Tn*(*f, P*) :=
*n*

X

*i=1*

*f*(*xi−*1) +*f*(*xi*)

*then*

*|Rn*(*f, P*)*|*=

Z *b*

*a*

*f*(*x*)*dx _{−}Tn*(

*f, P*)

*≤*

*K*
(*α*+ 2) (*α*+ 3)

*n*

X

*i=1*

*hα+2 _{i}*

*for some finite constantK >*0*.*

*Proof.* We just have to apply Theorem 2 to *f* on each subinterval [*xi−*1*, xi*]*,* for

*i*= 1*,*2*,*3*, ..., n.*This gives

*|Rn*(*f, P*)*|* =

Z *b*

*a*

*f*(*x*)*dx _{−}Tn*(

*f, P*)

=

*n*

X

*i=1*

Z *xi*

*xi−*1

*f*(*x*)*dx _{−}*

*n*

X

*i=1*

*f*(*xi−*1) +*f*(*xi*)

2 *hi*

=

*n*

X

*i=1*

*hi*

*f*(*xi−*1) +*f*(*xi*)

2 *−*

1
*hi*

Z *xi*

*xi−*1

*f*(*x*)*dx*!_{}

*≤*

*n*

X

*i=1*

*hi*

*f*(*xi−*1) +*f*(*xi*)

2 *−*

1
*hi*

Z *xi*

*xi−*1

*f*(*x*)*dx*
*≤*

*≤*

*n*

X

*i=1*

*hi*

*K*

(*α*+ 2) (*α*+ 3)(*xi−xi−*1)

*α+1*

= *K*

(*α*+ 2) (*α*+ 3)

*n*

X

*i=1*

*hα+2 _{i}*

*.*

and the theorem is completely proved.

The following three corollaries are natural consequences.

Corollary 2. *Assume that the hypotheses of Theorem 4 hold and thatP* *is a regular*
*partition of* [*a, b*]*,i.e.,* *xi*=*a*+*i* *b− _{n}a*

*fori*= 0*,*1*,*2*, ..., n.Then*

*|Rn*(*f, P*)*| ≤*

*K*(*b _{−}a*)

*α+2*(

*α*+ 2) (

*α*+ 3)

*nα+1*

*for some finite constantK >*0

*.*

*Moreover, if* *ε >*0 *is given, then*

*nε*:=

*K*(*b−a*)

*α+2*

(*α*+ 2) (*α*+ 3)*ε*
! 1

*α*+1
+ 1

*where, for* *r* _{∈}_{R}*,* [*r*] *is the integer part of* *r,* *is the smallest natural number for*
*which we have*

*|Rn*(*f*)*| ≤εforn≥nε.*

The second part of Corollary 2 says that if we use a regular (or uniform) partition
of [*a, b*] and we want the magnitude of the error to be less than some preassigned
tolerance*ε,*then we need to use at least

*nε*:=

*K*(*b−a*)

*α+2*

(*α*+ 2) (*α*+ 3)*ε*
! 1

partition points.

Corollary 3. *Let* *f* : [*a, b*]_{→}_{R}*be an absolutely continuous function such that* *f0*

*is Lipschitz continuous on* [*a, b*]*.* *If* *P* : *a* = *x*0 *< x*1 *< x*2 *< ... < xn* = *b* *is a*
*partition of the interval*[*a, b*]*, hi*=*xi−xi−*1 *fori*= 1*,*2*, ..., n,and*

*Tn*(*f, P*) :=
*n*

X

*i=1*

*f*(*xi−*1) +*f*(*xi*)

2 *·hi*

*Then*

*|Rn*(*f, P*)*| ≤*

*K*
12

*n*

X

*i=1*

*h*3_{i}.

*Moreover, the constant* 1

12 *is sharp.*

Corollary 4. *Assume that the hypotheses of Corollary 3 hold and that* *P* *is a*
*regular partition of* [*a, b*]*,* *i.e.,xi*=*a*+*i* *b− _{n}a*

*fori*= 0*,*1*,*2*, ..., n.Then*

*|Rn*(*f, P*)*|*=

Z *b*

*a*

*f*(*x*)*dx _{−}Tn*(

*f, P*)

*≤*

*L*(*b _{−}a*)3
12

*n*2

*.*

*Moreover, if* *ε >*0 *is given, then*

*nε*:=

s

*L*(*b _{−}a*)3
12

*ε*

+ 1

*is the smallest natural number for which we have*

*|Rn*(*f*)*| ≤εforn≥nε.*

In the case of functions which are twice differentiable and the second derivative is integrable, we have the following error estimate.

Theorem 5. *Letf* : [*a, b*]_{→}_{R}*be a twice differentiable function on*(*a, b*)*such that*

*f00 _{∈}_{L}*

_{1}

_{(}

_{a, b}_{)}

_{and}_{P}_{:}

_{a}_{=}

_{x}_{0}

_{< x}_{1}

_{< x}_{2}

_{< ... < x}_{n}_{=}

_{b}_{is a partition of the interval}[*a, b*]*.Then we have the following estimation*

*|Rn*(*f, P*)*|*

(3.1)

*≤* 1_{2}

*n*

X

*i=1*

1
*hi*

Z *xi*

*xi−*1
Z *xi*

*xi−*1

*K*1(*x, y*)*|x−y|dxdy*

*≤* *kPk*

2

6 *kf*

*00 _{k}*

1*,*

*whereK*1(*x, y*) =

R*y*

*x* *|f00*(*t*)*|dt*

*Proof.* Applying Theorem 3 on each subinterval [*xi−*1*, xi*] for *i* = 1*,*2*, ..., n,* we

obtain

*|Rn*(*f, P*)*|*

=

*n*

X

*i=1*

Z *xi*

*xi−*1

*f*(*x*)*dx _{−}*

*n*

X

*i=1*

*f*(*xi−*1) +*f*(*xi*)

2 *hi*

=

*n*

X

*i=1*

*hi*

1
*hi*

Z *xi*

*xi−*1

*f*(*x*)*dx _{−}f*(

*xi−*1) +

*f*(

*xi*) 2

!_{}

*≤*

*n*

X

*i=1*

*hi*

1
*hi*

Z *xi*

*xi−*1

*f*(*x*)*dx _{−}f*(

*xi−*1) +

*f*(

*xi*) 2

!_{}

*≤*

*n*

X

*i=1*

*hi*

1
2*h*2

*i*

Z *xi*

*xi−*1
Z *xi*

*xi−*1

*K*1(*x, y*)*|x−y|dxdy*

*≤*

*n*

X

*i=1*

*h*2
*i*

6
Z *xi*

*xi−*1

*|f0*(*t*)_{|}dt

*≤* *kPk*

2

6

*n*

X

*i=1*

Z *xi*

*xi−*1

*|f00*(*t*)_{|}dt

*≤* *kPk*

2

6 *kf*

*00 _{k}*

1*,*

which completes the proof.

Assume that the hypotheses of Theorem 5 hold and that*P* is a regular partition
of [*a, b*]*.*Then

*|Rn*(*f, P*)*| ≤*

(*b _{−}a*)2
6

*n*2

*kf*

*00 _{k}*

1*.*

If*ε >*0 is given, then

*nε*:=

"
(*b _{−}a*)

r

*kf00 _{k}*

1

6*ε*
#

+ 1

is the smallest natural number for which we have

*|Rn*(*f*)*| ≤ε*for*n≥nε.*

Finally, we have

Theorem 6. *Let* *f* : [*a, b*]_{→}R*be so that* *f0* _{is an absolutely continuous function}

*on* (*a, b*)*and such that* *f00* _{∈}_{L}_{p}_{(}_{a, b}_{)}_{,}_{where}_{1}_{< p <}_{∞}_{,}_{and}_{P}_{:}_{a}_{=}_{x}_{0} _{< x}_{1}_{<}

*x*2 *< ... < xn* =*b* *is a partition of the interval* [*a, b*]*.* *Then we have the following*
*estimation*

*|Rn*(*f, P*)*|*

*≤* 1_{2}

*n*

X

*i=1*

1
*hi*

Z *xi*

*xi−*1
Z *xi*

*xi−*1

*Kp*(*x, y*)*|x−y|*
2*−*1

*p _{dxdy}*

*≤* *p*

2

(3*p _{−}*1) (4

*p*1)

_{−}*n*

X

*i=1*

*h*
2*p−*1

*p−*1

*i*

!1*−*1

*p*

*where*

*Kp*(*x, y*) =

Z *y*

*x* *|*

*f00*(*t*)* _{|}pdt*
1

*p*

*, x, y _{∈}*[

*a, b*]

*.*

*The first inequality is sharp.*

A naturally corollary of this theorem is

Corollary 5. *Assume that the hypothesis of Theorem 6 hold and thatP* *is a regular*
*partition of* [*a, b*]*.Then*

*|Rn*(*f, P*)*| ≤*

*p*2(*b _{−}a*)2

*−p*1

*n*2_{(3}_{p}_{−}_{1) (4}_{p}_{−}_{1)}*kf*

*00 _{k}*

*p.*
*If* *ε >*0*is given, then*

*nε*:=

*p*

v u u

t(*b−a*)2*−*
1

*p _{k}_{f}00_{k}*

*p*

(3*p _{−}*1) (4

*p*1)

_{−}*ε* + 1

*is the smallest natural number for which we have*

*|Rn*(*f*)*| ≤εforn≥nε.*

References

[1] S. S. Dragomir, On the trapezoid inequality for absolutely continuous mappings,*submitted.*

[2] S. S. Dragomir, On the trapezoid formula for Lipschitzian mappings and applications,*Tamkang*
*Journal of Math. (in press)*

[3] S. S. Dragomir, On the trapezoid quadrature formula for mappings of bounded variation and
applications,*Extracta Mathematicae*,*(in press).*

[4] S. S. Dragomir, P. Cerone and A. Sofo, Some remarks on the trapezoid rule in numerical
integration,*submitted.*

[5] S. S. Dragomir and T. C. Peachey, New estimation of the remainder in the trapezoidal formula
with applications,*submitted.*

[6] S. S. Dragomir, P. Cerone and C. E. M Pearce, Generalization of the trapezoid inequality for
mappings of bounded variation and applications,*submitted.*

[7] N. S. Barnett, S. S. Dragomir, and C. E. M. Pearce, A quasi-trapezoid inequality for double
integrals*submitted.*

[8] S. S. Dragomir and A. McAndrew, On trapezoid inequality via a Gr¨uss type result and
appli-cations,*submitted.*

(S. S. Dragomir), School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City, MC 8001 Australia

*E-mail address*:sever@matilda.vu.edu.au

(S. Mabizela), Department of Mathematics and Applied Mathematics, University of Capetown, Rondebosch, 7700, SOUTH AFRICA