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- Homework Statement
- Calculate the area under a curve & provide the error term using 3 numerical methods

- Relevant Equations
- Midpoint, Trapezoidal & Simpson's rules

This isn't a homework question per se but I can post more details like the data points & my work after.

Suppose we are given a set of arbitrary points for which we cannot find an equation and we need to find the area under the curve without an analytical method - we can use either of the three numerical methods.

If we want to improve our area approximation, are we restricted to the number of points or is interpolation the only way to improve the result?

And finally, how do we calculate the error term? Would we use another numerical method to determine the second derivative in the case of midpoint & trapezoidal and the fourth derivative in the case of Simpson's rule?

Suppose we are given a set of arbitrary points for which we cannot find an equation and we need to find the area under the curve without an analytical method - we can use either of the three numerical methods.

If we want to improve our area approximation, are we restricted to the number of points or is interpolation the only way to improve the result?

And finally, how do we calculate the error term? Would we use another numerical method to determine the second derivative in the case of midpoint & trapezoidal and the fourth derivative in the case of Simpson's rule?