5. The Trapezoidal Rule

by M. Bourne

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:

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

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

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

Area of a trapezoid (trapezium)

`"Area"=h/2(p+q)`

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:

`Deltax=(b-a)/n`

and we also need

`y_0= f(a)`

`y_1= f(a + Δx)`

`y_2= f(a + 2Δx)`

`…`

`y_n= f(b)`

Note

`"Area"=int_a^b f(x)dx~~Deltax((y_0)/2+y_1+...+(y_n)/2)`

Exercise

Using `n= 5`, approximate the integral:

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

Didn't find what you are looking for on this page? Try search:

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Ready for a break?

 

Play a math game.

(Well, not really a math game, but each game was made using math...)

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Share IntMath!

Short URL for this Page

Save typing! You can use this URL to reach this page:

intmath.com/trapezoidal

Calculus Lessons on DVD

 

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

Loading...
Loading...