Course Homepage Preface On Learning
Build Your Intuition
1. 1-Minute Summary 2. X-Ray Vision 3. 3d Intuition
Learn The Lingo
4. Integrals, Derivatives 5. Computer Notation
Basic Understanding
6. Improved Algebra 7. Linear Changes 8. Squared Changes
Deeper Understanding
9. Infinity 10. Derivatives 11. Fundamental Theorem
Figure Out The Rules
12. Add, Multiply, Invert 13. Patterns In The Rules 14. Take Powers, Divide
Put It To Use
15. Archimedes' Formulas Summary
Welcome to an intuition-first calculus course.
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8 min read

5. Music From The Machine

In the previous lessons we've gradually sharpened our intuition:

Ring Detail

Wait! Our formal description is precise enough that a computer can do the work for us:

Wolfram Alpha

Whoa! We described our thoughts well enough that a computer did the legwork.

We didn't need to manually unroll the rings, draw the triangle, and find the area (which isn't overly tough in this case, but could have been). We saw what the steps would be, wrote them down, and fed them to a computer: boomshakala, we have the result. (Just worry about the "definite integral" portion for now.)

Now, how about deriviatives, X-Raying a pattern into steps? Well, we can ask for that too:

Wolfram Alpha

Similar to above, the computer X-Rayed the formula for area and split it step-by-step as it moved. The result is $$$2 \pi r$$$, the height of the ring at every position.

Seeing The Language In Action

Wolfram Alpha is an easy-to-use tool: the general format for calculus questions is

That's a little wordy. These shortcuts are closer to the math symbols:

Now that we have the machine handy, let's try a few of the results we've seen so far:

Formal (Computer) Description Math Notation Intuitive Notion
integrate 2 * pi * r * dr from r=0 to r=r $$\int_{0}^{r} 2 \pi r \ dr$$
integrate 1/2 * r * dp from p=0 to p=2*pi*r $$\int_0^{2 \pi r} \frac{1}{2} r \ dp$$
integrate 2 * sqrt(r^2 - x^2) from x = -r to x = r $$\int_{-r}^{r} 2 \sqrt{r^2 - x^2} \ dx$$

Click the formal description to see the computer crunch the numbers. As you might have expected, they all result in the familiar equation for area. A few notes:

The approach so far has been to immerse you in calculus thinking, and gradually introduce the notation. Some of it may be a whirl -- which is completely expected. You're sitting at a cafe, overhearing conversation in a foreign language.

Now that you have the sound in your head, we'll begin to explore the details piece-by-piece.


1) Are you getting a feel for what solving calculus problems feels like? What seems to make one approach "easier" to solve than another?

2) As always, any aha moments or questions?

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Class Discussion

Guided class discussions are available in the complete edition.