The modified trapezoidal rule for line integrals

Although the predictor-corrector curve tracing method has found a wide variety of significant applications, the calculation of line integrals over implicitly defined curves seems to have been overlooked. In this paper, we present two modified numerical integration techniques, namely, the modified trapezoidal and the modified Romberg rules and results of some numerical experiments for evaluating the line integral of a vector field over an implicitly defined curve. Moreover, we give a proof of an asymptotic error expansion for a modified trapezoidal rule which is a modification of that of Verlinden and Cools [8].