Integration and the Beauty of Wolfram Alpha

Back in the early 2000’s I had an evaluation copy of Wolfram Mathematica, which, don’t get me wrong, was great, but too clunky for me. I’m a hobbyist – love math, but easily confused. Also, the full featured product was like thousands of dollars, so – no way.

Fast forward a few years and I got an iPad and came across Wolfram Alpha and an app!! I think it was $2.99 – that’s my range 🙂

Well it has been just great, highly recommended. But I really didn’t go much past the obvious formulas, even though the integration was spectacular, with the ‘show me how you did that’ feature.

Recently I started reviewing my calculus. My son’s old books were just sitting around. Paying hundreds of dollars for his books, I figure I might as well get some use out of them, right?

So I am at the point in his calculus book where they are showing how volumes are computed. It’s fairly straightforward so far. This problem I am going to detail here gives you two functions (just based on x and y) and asks you to rotate the enclosed area around an axis and determine the volume that the structure holds. See the charts, it will be clearer.

I’ll include the problem here and some images, along with the solution in the book.

So my first thought was to calculate the area in the R region and then just revolve it around the x axis. I was able to get the area (1/6) – but revolving it doesn’t really work. (Multiply by 2*pi*R) I think the problem there is that the radius changes, but I have to think about it some more. Wolfram Alpha was easily able to tell me that 1/6 was the right answer for the area.

The point here is not the solution, but it is the fantastic ability for the Wolfram Alpha program to understand input. So let’s get on to that.

Just for the heck of it (and based on the instructions from Wolfram to just enter what you think will work), I entered the following, and the freaking app knew exactly what I meant:

As you can see it showed the method of integration needed. We are talking about a washer type shape – a larger circle of rotation with a smaller section cut out. (based on circular area formula pi*r^2)

If I had this back in the day, I probably would have graduated a grade point higher than I did! Stupid Slide Rule.

Not only does the app give you that solution, but a plethora of other information AND a graph:

Totally radical, dude.

Here’s another example. Two pages from the book and two images from Wolfram Alpha on my iPad.

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