# When do you use the trapezoidal rule?

One use is when the integrand does not have an antiderivative that is finitely expressible using familiar functions.

Many important (and interesting) functions do not have an antiderivative that can be written using a finite number of simpler functions.

An example you may be familiar with is the "bell curve" -- important in probability as relating to the normal distribution.

graph{y = e^(-1/2x^2)/sqrt(2pi) [-2.2, 2.126, -1.043, 1.121]}

The involves the integral:

So we need some kind of approximation method. We can use rectangles, but, in general, trapezoids give us a better approximation with the same number of arithmetic steps.

For even better approximations we can use Simpson's rule (using parabolas).

And there are other ways to approximate. The point is that sometimes we just have to use an approximation technique and the trapezoidal rule can be explained in a first calculus course.

In addition to the area mentioned above, natural logarithms are calculated by approximation techniques.

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The trapezoidal rule is used to approximate the definite integral of a function when the function is difficult or impossible to integrate analytically. It is particularly useful when the function is given by a set of discrete data points rather than a continuous equation. The trapezoidal rule is often applied in numerical integration when a more accurate method, such as Simpson's rule, is not required or feasible.

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