Numerical analysis 101. The trapezoidal rule, when applied to a FIXED sequence of data points, will return a SINGLE estimate of the area contained. This estimate is based on the area underneath a piecewise linear function that in turn defines a set of trapezoids. Compute the area of the trapezoids, and form the sum. Thus is the origin of the name - trapezoidal rule. There is no control over a convergence tolerance here. None at all.
Yes, there are other methods. You might choose to read about other Newton-Cotes rules. Simpson's rule is one. If you look on the FEX, you will find Simpson's rule codes. I don't believe that you will find any higher order codes than this on the FEX. (Be careful, as Simpson's rule has in it an implicit requirement on the number of points used. This is true of any Newton-Cotes rule beyond trapezoidal rule. Some of those tools on the FEX forget to mention they have this flaw, or fail to deal with it creatively.)
Note that you have no control over the error tolerance for Simpson's rule either. It MAY often be more accurate than trapezoidal rule, however, I can show circumstances where too high of an order integration rule will degrade your results. Usually this problem results in the case of noise. If your data has noise in it, then the noise will become amplified. Of course, when working with floating point arithmetic, noise always enters your data, but it is really tiny noise, down in the least significant bits of your numbers.
Finally, nothing stops you from writing code for a higher order Newton-Cotes rule. Go too high in the order, and you may well be wasting your time. It depends on your data, the source of it, etc.
