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In my calculus class, we learned about using Simpson's Rule to approximate the value of an integral. Since it uses parabolas instead of lines to approximate the integral, it becomes more accurate than the trapezoid or midpoint rules.

My question is this: why do people stop at quadratics? Shouldn't it be more accurate to use even higher degree polynomials?

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    Higher degree polynomials can be used, but they tend to have more and more oscillations in them when you attempt to build a function out of them – 2017-01-25
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    A meta-reason for this is that it is still feasible for calculus students to apply Simpson's rule to approximate a given integral by hand. The higher-order methods grow more and more tedious :) – 2017-01-25

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There are rules higher up than Simpson's rule, known as Boole's rule and Weddle's rule, and people do use them for numerical integration. Here's an example of the use of Weddle's rule in nondestructive testing, pointing out that Weddle's rule is exact for fifth degree polynomials.

These are all higher-order versions of Newton-Cotes formulas. The Wikipedia article notes that higher-order versions can suffer from the catastrophic Runge's phenomenon: going to higher degree interpolation does not always improve accuracy.

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The end of a subject in your calculus course is not the end of that subject in mathematics. There is a lot more to numerical integration. Yes, higher-order methods can be, and are, used.