The Improvement of the Numerical Integral Formula Based on the Taylor Formula

By using Taylor’s formula for function at the end point and middle point, a kind of remainder term formula about the rectangular quadrature formula of the numerical integral formula can be obtained. Based on the fixed value of the remainder formula at a certain point, proper modification is made and the improved quadrature formula for left, right and middle rectangle is worked out. The higher the order of the Taylor’s formula is, the higher of the algebraic accuracy of the improved rectangular quadrature formula will be. The improved trapezium quadrature formula is then obtained through the improved rectangular quadrature formula. And the computational example is conducted to prove its validity.