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.

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.

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Answer 2

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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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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