The Trapezoidal Rule

Problem: Find

We put u = x² + 1 and as x = 0 → 1,u = 1 → 2

If u = x² + 1 then du = 2x dx.

But the question does not contain an x dxterm so we cannot solve it using any of the normal integration methods.

We need to use numerical approaches. When software like Mathcad or graphics calculators perform definite integrals, they use numerical methods given earlier.

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:

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 ≈

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

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

and we also need

y₀ = f(a)

y₁ = f(a + Δx)

y₂ = f(a + 2Δx)
…

y_n = f(b)

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Note:

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

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Exercise: Using n = 5, approximate the integral:

Answer

Here, a = 0 and b = 1.

y₀ = f(a) = f(0) =  = 1
y₁ = f(a + Δx) = f(0.2) = 
y₂ = f(a + 2Δx) = f(0.4) = 
y₃ = f(a + 3Δx) = f(0.6) = 
y₄ = f(a + 4Δx) = f(0.8) = 
y₅ = f(b) = f(1) = 
So the area ≈

So  ≈1.150