Imagine you have an x-y graph with a plotted line running from left to right. In addition to marking these specific data points, you may need to calculate the area of the region under the line. While this is easy enough to do with a straight line, a curving, irregular line is more problematic. This is where the trapezoidal rule comes into play.
The trapezoidal rule creates a series of side-by-side, left-to-right trapezoids under the curve. All the individual trapezoid areas are then added together to calculate the total area under the x-y points making up the curving line. The more trapezoids you use, the more accurate the total area calculated will be.
If you’re taking calculus or are an engineer, you likely use the trapezoidal rule on a regular basis. And, while you can solve this by hand, there’s no need to because of the free trapezoidal rule calculators and software available to do it for you. We’ll go over your best options below.
TRAPEZOIDAL RULE APPLICATIONS
The first known use of the trapezoidal rule dates to 50 BCE when it was used for integrating Jupiter’s velocity on the ecliptic. While other equations such as Simpson’s Rule can provide an even more accurate integral – that is, the total area under the graph – the trapezoidal rule is still used for periodic functions and double exponential functions.
In more tangible, real-world applications, the trapezoidal rule is often used in civil engineering:
- Calculating volume from cross-sections: The amount of earth to be excavated for building foundations or for a reservoir.
- Calculating earth work from spot levels: The amount of earth needing to be moved for proper elevation of basements.
- Calculating volume from contours: The amount of water necessary to fill a reservoir.
BEST FREE TRAPEZOIDAL RULE CALCULATORS
If you need a free trapezoidal rule calculator, there are websites which will do this for you while providing varying degrees of information. You can also download software to your graphing calculator or computer. Here are your best options.
Wolfram|Alpha’s goal is to make systematic knowledge computable and accessible to everyone. This has been made possible over the past thirty years via two means: the Wolfram Language programming code and a New Kind of Science (NKS), the discovery of new algorithms for computation. Designed to be used by students, educators, and researchers, a variety of online and mobile apps are available.
On the trapezoidal rule calculator page, there are four input fields: function, number of trapezoids, lower limit, and upper limit. For example, you could input the following information:
- Function: sin(x^2)
- Number of trapezoids: 8
- Lower limit: 0
- Upper limit: 2
After pressing Submit, you’ll get the answer: 0.790821. You will not, however, receive any additional information about how the calculation was performed.
You can share the page via email, Twitter, and Facebook. It’s also easy to customize the calculator and embed it on your own webpage, Blogger, or WordPress site. In addition, if you set up an account at Wolfram|Alpha, you can build your own apps through the developer interface.
This site is more basic and less robust in its offerings than Wolfram|Alpha, yet it does provide some additional information of value to students in particular. Once again, to use the trapezoidal rule calculator, you’ll be prompted to enter the same four categories of information as above albeit in a slightly different order:
- Function: sqrt (2+ sin ^3 (x))
- Number of trapezoids: 0
- Lower limit: 1
- Upper limit: 8
After pressing Submit, you’ll get the answer: 1.4753289483117. Unlike Wolfram|Alpha, however, you have the option to see how all the steps were performed in addition to accompanying explanatory information.
Unfortunately, there is no option to embed this calculator on your own site even though you can share it via email or social media. There are also quite a few advertisements which clutter the page’s layout.
Based in Russia, PlanetCalc is a collection of more than 550 online calculators. If you need a calculator not on the site, you can submit a detailed request which will be directed to one of its more than 100 volunteer programmers. Any PlanetCalc calculator can also be embedded on your website.
As with the previous online calculators, you’ll enter the function, lower and upper limits, and the number of intervals/trapezoids. Unlike them, however, you can adjust the precision of the answer from zero to twenty decimal points. You will also be able to see all the steps performed in the calculation as well as accompanying explanatory notes. Even more helpful, an x-y graph as per your inputted data will be generated so you’ll have a visual representation of what you’ve just calculated.
GeoGebra is another education site with many online apps for everything from geometry to 3D graphing to spreadsheets and more. In addition, many of the apps can be downloaded for iOS, Android, Windows, Mac, Chromebook, and Linux operating systems.
What makes GeoGebra’s trapezoidal rule calculator stand out is its interactive, split-screen design. On the left-hand side, you’re prompted to enter the necessary information to perform integral calculation. On the right-hand side, there’s a live x-y axis which immediately adjusts your graph with the data you’ve inputted. Plus, it shows you the steps the solve the equation.
This interactive design is also available for other, even more complex online calculators including the Simpson Rule of Numerical Integration as well as a comparison of the trapezoidal rule and Simpson Rule which allows you to toggle back and forth between the two.
NA-Labs is another barebones site which offers five online calculators: the trapezoidal rule, Simpson’s Rule, Riemann Right Endpoint Rule, Riemann Left Endpoint Rule, and Riemann Midpoint Rule.
Once again, you’ll enter the function, upper and lower limits, and the number of trapezoids to use. The default results are displayed in degrees mode. You are given the option, however, to easily convert to radians for evaluating trigonometric functions.
WanerMath is another stripped-down site which nonetheless provides a number of useful resources for finite mathematics and applied calculus for students. These include online tutorials, review exercises, and true-false quizzes. In addition, the site is available in Spanish.
Instead of offering an online calculator like the options above, the trapezoidal rule webpage provides the step-by-step code to program your TI-83 graphing calculator to do this for you. After all, a webpage is handy enough, but being able to perform multiple trapezoidal rule calculations without needing Internet access is a big plus. You can also immediately use the data you generate for other equations and operations on your calculator.
The coding is relatively simple as there are only 20 lines required to compute left- and right-hand Riemann sums using the trapezoidal rule. While you’re there, you can also grab the code to use Simpson’s Rule, a 17-line program. You can’t beat it: A few lines of code to perform heavy-duty calculations any time you have the need.
One of the oldest, unofficial Texas Instruments support sites, ticalc.org is loaded with useful information. Registered users can upload and download files, read and submit reviews, and rate files based on their quality. You’ll also be able to access new games for TI calculators as well as patches and other fixes.
This site is useful because you may not want to manually key in the code on your TI-83 or TI-84 Plus. On ticalc.org’s TI-83/84 Plus Basic Math Programs (Calculus) page, you can access almost 300 program files to download to your TI graphing calculator. This includes over 25 different trapezoidal rule calculator programs. In addition to the trapezoidal rule, many of these files contain multiple calculus-related programs you’ll find handy too.
There is also a handy legend which indicates which features a file has: file with screenshots, file with animated screenshots, and file with reviews.
Then again, perhaps you want the software for a trapezoidal rule calculator to run on a machine other than a TI calculator. In that case, CodingAlpha is a site you should look at as it provides the code to do this in the C programming language.
In less than 30 lines of C programming code, you’ll be able to calculate the trapezoidal rule on your own computer. Not only is the code provided, but there are handy troubleshooting tips too. The code utilized a GNU GGC compiler on a Linux Ubuntu operating system, but it should run on any other operating system too.
Users can also download C programs for other numerical calculations such as Weddle’s Rule Algorithm, Picard’s Method, and Euler’s Method.
USING TRAPEZOIDAL RULE CALCULATORS
Just like engineers don’t use slide rules anymore, there’s no need to calculate trapezoidal rule integrals by hand. Whether you want to use an online calculator or download code to your own graphing calculator or computer, there are many options available to let you work smart, not hard. That way, you can spend more time on the big picture issues of the problem you are working to solve.
Need to calculate arc length? Check out our how-to guide to perform this important equation!