{"id":4790,"date":"2019-11-12T17:30:52","date_gmt":"2019-11-12T22:30:52","guid":{"rendered":"http:\/\/baselines.com\/?p=4790"},"modified":"2019-11-14T11:16:26","modified_gmt":"2019-11-14T16:16:26","slug":"integration-and-the-beauty-of-wolfram-alpha","status":"publish","type":"post","link":"https:\/\/baselines.com\/?p=4790","title":{"rendered":"Integration and the Beauty of Wolfram Alpha"},"content":{"rendered":"\n

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.<\/p>\n\n\n\n

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 \ud83d\ude42<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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?<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

I’ll include the problem here and some images, along with the solution in the book.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

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.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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)<\/p>\n\n\n\n

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

Not only does the app give you that solution, but a plethora of other information AND a graph:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Totally radical, dude. <\/p>\n\n\n\n

Here’s another example. Two pages from the book and two images from Wolfram Alpha on my iPad.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n