The Trapezoidal Rule

You have likely already read our Forming Riemann Sums article, looking at how we can use them to estimate the area under the curve. As a refresher, a Riemann sum divides the area under the curve into rectangles and sums the area of each rectangle to approximate the area under the curve. But do we always have to use rectangles when approximating areas? Of course not!

The Trapezoidal Rule is a form of Riemann's sum. However, the Trapezoidal Rule uses trapezoids rather than rectangles! Interestingly enough, using trapezoidal subregions to approximate the area is usually more exact than using rectangles. In this article, we'll explore the Trapezoidal Rule derivation, formula, and error. Finally, we'll apply the Trapezoidal Rule to some examples.

The Trapezoidal Rule Definition and Formula for Area

Before we get into how this technique is used in practice, let's define what this rule is!

The Trapezoidal Rule estimates the area under the curve by dividing the region into trapezoidal subregions - StudySmarter Original

The area of a trapezoid is defined as

$\frac{1}{2}\times (\mathrm{distance}\mathrm{between}\mathrm{each}\mathrm{base})\times (\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{each}\mathrm{base})$

Translating this formula to the above figure, we can say that the area of the left-most trapezoid is defined as

$\frac{1}{2}\times (∆x)\times \left(f\right(x_0)+f(x_1\left)\right)=\frac{∆x}{2}\times \left(f\left(x_0\right)+f\left(x_1\right)\right)$

Similarly, the formula for the area of the 2nd left-most trapezoid is defined as

$\frac{1}{2}\times \left(∆x\right)\times \left(f\left(x_1\right)+f\left(x_2\right)\right)=\frac{∆x}{2}\times \left(f\left(x_1\right)+f\left(x_2\right)\right)$

The formula for the area of each trapezoid is formulated the same way. The Trapezoidal Rule states that we can estimate the area under the curve by summing the area of each of these trapezoids. The Trapezoidal Rule is derived by factoring out $\frac{∆x}{2}$ and adding up the length of each base, where $f\left(x_1\right)$ through $f\left(x_{n-1}\right)$ are multiplied by a factor of two because they are bases shared by other trapezoids.

Then, for approximating the definite integral of a function f(x), the Trapezoidal Rule states

where n is the number of trapezoids, $∆x=\frac{b-a}{n}$, and $x_i=a+i∆x$.

Over and Underestimating using the Trapezoidal Rule

Look at the graph under the Trapezoidal Rule definition again. Notice how some of the trapezoidal subregions stay under the graph while other subregions stick out over the graph. When the graph is "concave up" (the graph bends upwards), the subregions tend to overestimate the area under the curve.

When the graph is "concave down" (the graph bends downwards), the subregions tend to underestimate the area under the curve. Based on a function's concavity, we can use this observation to tell whether the Trapezoidal Rule will overestimate or underestimate the area under the curve.

Below is a graphical example illustrating the difference between an overestimate and an underestimate.

Notice how the trapezoids extend above the function when there is an overestimation and sit below the function when there is an underestimation - StudySmarter Originals

The Trapezoidal Rule Error Bounds

As numerical integration techniques, like the Trapezoidal Rule, are an estimation, calculating the error of that estimation is incredibly important.

Relative error

Using common sense, we compute the relative error of a Trapezoidal Rule computation (given as a percentage) by using the relative error formula:

where $T_n$ is the Trapezoidal Rule approximation of the integral and ${\int }_a^{b}f\left(x\right)dx$ is the actual area.

Absolute Error

In addition to relative error, the absolute error of our approximation using the trapezoidal rule can be calculated using the formula for absolute error:

$\begin{array}{rcl}Absoluteerror& =& \left|approximation-actual\right|\\ & =& \left|T_n-{\int }_a^{b}f\left(x\right)dx\right|\end{array}$

Error Bounds for the Trapezoidal Rule

We can use an error bound formula to tell us the maximum possible area of our approximation. For the Trapezoidal Rule, the error bound formula is

$\left|E_T\right|\le \frac{K{\left(b-a\right)}^3}{12n^2}$ for $\left|f\text{'}\text{'}\left(x\right)\right|\le K$

where $E_T$ is the exact error for the Trapezoidal Rule and $f\text{'}\text{'}\left(x\right)$ is the second derivative of f(x). Essentially, K is the maximum value of the second derivative on the interval [a, b].

The uses of the error bound will make more sense once we work through some examples.

Examples of Using the Trapezoidal Rule to Estimate the Integral

Example 1

Trapezoidal Rule vs. Simpson's Rule

Estimating areas using trapezoids and rectangles use straight lines on top of the shape. When reading the article on Simpson's Rule, you'll discover that we replace the straight lines in the trapezoids and rectangles with a curve (more specifically, a parabolic curve). More on this in the Simpson's Rule article!