5. The Trapezoidal Rule

by M. Bourne

Interactive exploration

See an applet where you can explore Simpson's Rule and other numerical techniques:

Riemann Sums Applet

Problem: Find

`int_0^1sqrt(x^2+1)\ dx`

We put `u = x^2+ 1` then `du = 2x\ dx`.

But the question does not contain an `x\ dx` term so we cannot solve it using any of the integration methods we have met so far.

We need to use numerical approaches. (This is usually how software like Mathcad or graphics calculators perform definite integrals).

We can use one of two methods:

Continues below

The Trapezoidal Rule

We saw the basic idea in our first attempt at solving the area under the arches problem earlier.

Instead of using rectangles as we did in the arches problem, we'll use trapezoids (trapeziums) and we'll find that it gives a better approximation to the area.

Area under curve using trapezoidal Rule

Area under curve using trapezoidal Rule

Recall that we write "Δx" to mean "a small change in x".

Area of a trapezoid (trapezium)

Now, the area of a trapezoid (trapezium) is given by:


So the approximate area under the curve is found by adding the area of the trapezoids. (Our trapezoids are rotated 90° so that their new base is actually the height. So h = Δx.)

`"Area"~~1/2(y_0+y_1)Deltax+` `1/2(y_1+y_2)Deltax+` `1/2(y_2+y_3)Deltax+...`

We can simplify this to give us the Trapezoidal Rule, for `n` trapezoids:

`"Area"~~` `Deltax((y_0)/2+y_1+y_2+y_3+` `{:...+(y_n)/2)`

To find `Δx` for the area from `x=a` to `x=b`, we use:


and we also need

`y_0= f(a)`

`y_1= f(a + Δx)`

`y_2= f(a + 2Δx)`


`y_n= f(b)`


  • We get a better approximation if we take more trapezoids [up to a limit!].
  • The more trapezoids we take, `Delta x` will tend to `0`, that is, `Δx → 0.`
  • We can write (if the curve is above the x-axis only between `x=a` and `x=b`):

`"Area"=int_a^b f(x)dx`

`~~` `Deltax((y_0)/2+y_1+{:...+(y_n)/2)`


Using `n= 5`, approximate the integral:

`int_0^1sqrt(x^2+1)\ dx`


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