For Calculus – Integrate Anything With Wolfram
I took calculus in high school, and I used to know it pretty well. Unfortunately, I’ve gotten a bit rusty over the last ten years. So when some of my students ask me for help with calculus I sometimes get confused…
And that’s where Wolfram Mathematica’s Online Integrator comes in.
I love this little tool. You can type in any function you want, and it will integrate it for you. It’s an online web application built with Wolfram’s Mathematica – a framework capable of doing some really complicated calculations. I’ve toyed around with programming various types of calculators and simple calculations, like factoring, and I can’t even imagine how complicated it would be to program this calculator from scratch.
But, you don’t have to. The beauty is the simplicity. Type in the most complicated, obscure integration, and Mathematica will spit out an answer for you.
So You Want Kids to Cheat and Take the Easy Way Out?
Not so much. If you’re learning calc, then you need to learn to integrate functions. There’s no way around it. Moreover, you need to understand the logic and theory underlying the relationship between integrals and derivatives.
Once you’ve mastered that relationship, though, you don’t necessarily have to work out every integration by hand any more than you have to work out every bit of division by hand. We use calculators as a way to improve efficiency at every level of math, and it’s up to the teacher (and the student) to decide when a certain skill has been mastered and a calculator can be used to supplement the student’s brain.
My dad was an engineering student at Ohio State in the late 60’s and early 70’s. He’s told the story many times about how he bought his first calculator. Texas Instruments had introduced the first basic (four function, I think) calculator in the late ’60’s. They were ridiculously expensive. After puzzling over a problem with his slide rule for hours, he decided to bite the bullet and drop a couple hundred bucks on a calculator. A few minutes later, the problem was solved.
Why spend hours puzzling over something that we have the tools to solve much more quickly? There’s a reason we invented calculators. And there’s a reason we let kids learn how to use them.
So How Would You Use It?
Well, I wouldn’t introduce it on the first day of class. I’d first teach students the basics of integration and try to build an understanding of the relationship between functions that is represented by integrals and derivatives.
Once students have mastered the basic mechanics of integration, however, I would show them the Online Integrator for three reasons.
Instant Feedback. Students need feedback, and the quicker the better. With the online integrator, students can instantly run an equation through the calculator, get an accurate answer, and compare it to their work. Worried that students will just punch things into the calculator? Make them show their work.
Confused? Use the Answer to Help. Sometimes a complicated integral is just downright obscure and confusing. You might not understand what trig identity your supposed to use or how you’re supposed to factor the polynomials involved. Having an answer is a way to scaffold these complex problems. Students know where they’re starting and they know where they’re ending up. They need to work out the steps to get from point A to point B.
Efficiency for Complex Problems. In later stages of calculus, you get into more complex problems like the volume of a shape created by rotating a function around an axis. At this point, students know how to integrate functions; they are trying to master the skill of setting up an integrand that properly represents the shape. Using the online integrator allows students to focus on the task at hand, rather than expending all their brainpower on integrating functions.
What Do You Think?
Of course, I’m just a lowly history teacher. What do I know about math.
I’d be curious to hear from some high school calc teachers… what do you think about the online integrator? Useful tool, or a crutch that prevents students from ever really learning calculus?