Question

When do you use the trapezoidal rule?

Answer 1

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

Explanation:

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]}

To find probability we often need to find the area under this curve from some `x=a` to `x=b`

The involves the integral:

`int_a^b e^(1/2x^2) dx`

The function `e^(1/2x^2)` does not have a 'nicely' expressible antiderivative, so we cannot use the Fundamental Theorem of Calculus.
(As we could, for example to find `int_a^b (x^3+x^(5/7))dx`.)

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.