# Formula(s) for composite/multiple-segment trapezoidal rule

• s3a
In summary, the conversation discussed different formulas for the composite/multiple-segment trapezoidal rule used to approximate the value of an integral. One source suggests using N-1 while another suggests using N as the upper limit of the summation, and a third source includes an additional term. The speaker believes the first formula to be the correct one and suggests testing it with a simple example.
s3a

## Homework Statement

Hello, everyone. :)

I'm looking at several different sources for the composite/multiple-segment trapezoidal rule (for approximating the value of an integral), but the formulas don't seem to agree.

## Homework Equations

One place says I ≈ Δx/2 [f(a) + f(b)] + Δx Σ_(i=2)^(N-1) f(x_i).

Another place says I ≈ Δx/2 [f(a) + f(b)] + Δx Σ_(i=2)^(N) f(x_i).

Yet another place says I ≈ Δx/2 [f(a) + f(b) + 2 Σ_(i=1)^(N-1) f(a + iΔx)].

## The Attempt at a Solution

Only the first one seems correct to me. Basically, the second one seems to me like it should end at N-1, not N, and the third one seems to me like it should end at N-2, not N-1.

Am I right?

You could set up a simple example, like if you are splitting your integral into 3 trapezoids. Then see how each of these formulas plays out in figuring it.

## 1. What is the formula for the composite trapezoidal rule?

The formula for the composite trapezoidal rule is given by:
ab f(x) dx ≈ h/2 * [f(a) + 2*f(x1) + 2*f(x2) + ... + 2*f(xn-1) + f(b)]
where h = (b-a)/n and n is the number of subintervals.

## 2. How is the composite trapezoidal rule different from the regular trapezoidal rule?

The composite trapezoidal rule is an extension of the regular trapezoidal rule. It uses multiple trapezoids to approximate the area under a curve, whereas the regular trapezoidal rule only uses one trapezoid. This makes the composite rule more accurate for functions that are not well approximated by a single trapezoid.

## 3. What is the formula for the multiple-segment trapezoidal rule?

The formula for the multiple-segment trapezoidal rule is given by:
ab f(x) dx ≈ h/2 * [f(a) + 2*f(x1) + 2*f(x2) + ... + 2*f(xn-1) + f(b)]
where h = (b-a)/n and n is the number of segments used.

## 4. When should I use the composite trapezoidal rule instead of the multiple-segment trapezoidal rule?

The composite trapezoidal rule is more suitable for functions that are not well approximated by a single trapezoid, while the multiple-segment trapezoidal rule is more suitable for functions that can be approximated well by a smaller number of segments. Therefore, if the function is more complex, it is recommended to use the composite trapezoidal rule for better accuracy.

## 5. Can the composite trapezoidal rule be used to approximate definite integrals with infinite bounds?

No, the composite trapezoidal rule can only be used for definite integrals with finite bounds. If the bounds are infinite, other methods such as the Simpson's rule or the Gaussian quadrature method should be used.

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