Asian Journal of Mathematics and Computer Research

This paper presents a closed mixed quadrature rule constructed by averaging the Lobatto three-point rule and an anti-Lobatto four-point rule derived from the Lobatto formulation. The proposed rule is developed on the standard interval and is designed to improve the degree of precision obtained from its two constituent rules without changing the closed-type structure of the approximation. The construction combines two rules of precision three so that the resulting averaged Lobatto rule attains precision five. The truncation error is examined through series expansions and compared with the error terms of the base rules. The analysis indicates that the leading error term of the proposed rule is of higher order than those of the individual Lobatto and anti-Lobatto rules under the required smoothness assumptions. An adaptive quadrature routine using the proposed rule as the core approximation formula is also considered. To assess the numerical behaviour of the method, five definite test integrals with known exact values are evaluated using the Lobatto rule, the anti-Lobatto rule, and the proposed averaged rule. The numerical results show that the averaged rule gives smaller absolute errors than the two constituent rules for the selected test cases. The adaptive implementation further reduces the number of subintervals required to achieve the prescribed tolerance in the worked-out examples. These findings indicate that the proposed averaged Lobatto quadrature rule can serve as an effective approach for approximating definite integrals when increased precision is required within a closed-type quadrature framework. The study is limited to the theoretical derivation and numerical verification of the proposed rule using the selected test integrals.