5. The Trapezoidal Rule

by M. Bourne

Problem: Find

math expression

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

So du = 2x dx

But the question does not contain an x dx term 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.

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 above.

Instead of using rectangles, we see that trapezoids (trapeziums) give a better approximation to the area.

math expression


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:

math expression

math expression

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 ≈

math expression

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

math expression

[This is less calculation than the form used in the text. You may use either.]


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

math expression

and we also need

y0 = f(a)

y1 = f(a + Δx)

y2 = f(a +x)

yn = f(b)


Note:

math expression


Exercise: Using n = 5, approximate

math expression

 

Here, a = 0 and b = 1.

math expression

y0 = f(a) = f(0) = math expression = 1

y1 = f(a + Δx) = f(0.2) = math expression

y2 = f(a +x) = f(0.4) = math expression

y3 = f(a +x) = f(0.6) = math expression

y4 = f(a +x) = f(0.8) = math expression

y5 = f(b) = f(1) = math expression

So the area ≈

math expression


So math expression 1.150



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