Chapter Contents

• Integration
• 1. The Differential
• 2. Antiderivatives and The Indefinite Integral
• 3. The Area Under a Curve
• 4. The Definite Integral
• 5. Trapezoidal Rule
• 6. Simpson’s Rule
• Riemann Sums
• Integration Mini-lectures
• The Differential
• Difference Between Differentiation and Integration
• Given dy/dx, find y = f(x)
• Integration by Substitution
• Difference Between Definite and Indefinite Integrals
• Area Under a Curve
• Millionaire Calculus Game
• Integration Problem Solver

• Comments, Questions?

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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:

• Trapezoidal rule
• Simpson's Rule (in the next section: 6. Simpson's Rule)

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.

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

Let's see this in LiveMath. Note that our approximation is much better than using rectangles.

LIVEMath

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

`"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

• 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)`

Exercise

Using `n= 5`, approximate the integral:

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

Answer