A list of clickable links, by chapter.
- integral – http://www.wolframalpha.com/input/?i=integrate+2+*+pi+*+r+*+dr+from+r%3D0+to+r%3D0.5r
- integral – http://www.wolframalpha.com/input/?i=integrate+2+*+pi+*+r+*+dr+from+r%3D0.5r+to+r%3Dr
- description – http://www.wolframalpha.com/input/?i=integrate+1%2F2+*+r+*+dp+from+p%3D0+to+p%3D2*pi*r
- area – http://www.wolframalpha.com/input/?i=derive+4%2F3+pi+*+r%5E3+with+respect+to+r
- do the work for us – http://www.wolframalpha.com/input/?i=integrate+2+*+pi+*+r+*+dr+from+r%3D0+to+r%3Dr
- Now, there may be piles he’s never seen, that are difficult or impossible to recognize.
- Here’s another example: can you divide a cake into 3 equal portions, by only cutting into quarters?
- (Still shaky about exactly how dx can appear and disappear? You’re in good company. This question took top mathematicians decades to resolve. Here’s a deeper discussion of how the theory works.)
- You can check your answers with Wolfram Alpha, such as
And by the Pythagorean theorem, we have a connection between the x-position of the plate, and its height.
Think that was hard work? You have no idea. That one-line computation took Archimedes, one of the greatest geniuses of all time, tremendous effort to figure out. He had to imagine some spheres, and a cylinder, and some cones, and a fulcrum, and imagine them balancing and… let’s just say when he found the formula, he had it written on his grave. Your current intuition would have saved him incredible effort (see this video).
- Wow, that was fast! The order of our morph (Circumference → Area → Volume → Surface area) made the last step simple. We could try to spin a circumference into surface area directly, but it’s more complex.
Elementary Calculus: An Infinitesimal Approach by Jerome Keisler (2002). This book is based on infinitesimals (an alternative to limits, which I like) and has plenty of practice problems. Available in print or free online.
Calculus Made Easy by Silvanus Thompson (1914). This book follows the traditional limit approach, and is written in a down-to-earth style. Available on Project Gutenberg and print.
MIT 1801: Single Variable Calculus. Includes video lectures, assignments, exams, and solutions. Available free online.