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.
Read online, or buy the complete edition for videos, PDF, and discussion.
12 min read

4. Learning The Official Terms

We've been able to describe our step-by-step process with analogies (X-Rays, Time-lapses, and rings) and diagrams:

Circles and spheres

However, this is a very elaborate way to communicate. Here's the Official Math® terms:

Intuitive Concept Formal Name Symbol
X-Ray (split apart) Take the derivative (derive) $$\frac{d}{dr}$$
Time-lapse (glue together) Take the integral (integrate) $$ \int $$
Arrow direction Integrate or derive "with respect to" a variable. $$$dr$$$ implies moving along $$$r$$$
Arrow start/stop Bounds or range of integration $$\int_{start}^{end} $$
Slice Integrand (shape being glued together, such as a ring) Equation, such as $$$2 \pi r$$$

Let's walk through the fancy names.

The Derivative

The derivative is splitting a shape into sections as we move along a path (i.e., X-Raying it). Now here's the trick: although the derivative generates the entire sequence of sections (the black line), we can also extract a single one.

Think about a function like $$$f(x) = x^2$$$. It's a curve that describes a giant list of possibilities (1, 4, 9, 16, 25, etc.). We can graph the entire curve, sure, or examine the value of f(x) at a specific value, like $$$x = 3$$$.

The derivative is similar. Officially, it's the entire pattern of sections, but we can zero into a specific one by asking for the derivative at a certain value. (The derivative is a function, just like $$$f(x) = x^2$$$; if not otherwise specified, we're describing the entire function.)

What do we need to find the derivative? The shape to split apart, and the path to follow as we cut it up (the orange arrow). For example:

I agree that "with respect to" sounds formal: Honorable Grand Poombah radius, it is with respect to you that we derive. Math is a gentleman's game, I suppose.

Taking the derivative is also called "differentiating", because we are finding the difference between successive positions as a shape grows. (As we grow the radius of a circle, the difference between the current disc and the next size up is that outer ring.)

The Integral, Arrows, and Slices

The integral is glueing together (time-lapsing) a group of sections and measuring the final result. For example, we glued together the rings (into a "ring triangle") and saw it accumulated to $$$\pi r^2$$$, aka the area of a circle.

Here's what we need to find the integral:

Nope! We need to be specific. We've been saying we cut a circle into "rings" or "pizza slices" or "boards". But that's not specific enough; it's like a BBQ recipe that says "Cook meat. Flavor to taste."

Maybe an expert knows what to do, but we need more specifics. How large, exactly, is each step (technically called the "integrand")?

Circles and spheres

Ah. A few notes about the variables:

There's several gotchas to keep in mind.

First, $$$dr$$$ is its own variable, and not "d times r". It represents the tiny section of the radius present in the current step. This symbol ($$$dr$$$, $$$dx$$$, etc.) is often separated from the integrand by just a space, and it's assumed to be multiplied (written $$$2 \pi r \ dr$$$).

Next, if $$$r$$$ is the only variable used in the integral, then $$$dr$$$ is assumed to be there. So if you see $$$\int 2 \pi r$$$ this still implies we're doing the full $$$\int 2 \pi r \ dr$$$. (Again, if there are two variables involved, like radius and perimeter, you need to clarify which step we're using: $$$dr$$$ or $$$dp$$$?)

Last, remember that $$$r$$$ (the radius) changes as we time-lapse, starting at 0 and eventually reaching its final value. When we see $$$r$$$ in the context of a step, it means "the size of the radius at the current step" and not the final value it may ultimately have.

These issues are extremely confusing. I'd prefer we use rdr to indicate an intermediate "r at the current step" instead of a general-purpose "r" that's easily confused with the max value of the radius. I can't change the symbols at this point, unfortunately.

Practicing The Lingo

Let's learn to talk like calculus natives. Here's how we can describe our X-Ray strategies:

Intuitive Visualization Formal description Symbol
derive the area of a circle with respect to the radius $$\frac{d}{dr} Area $$
derive the area of a circle with respect to the perimeter $$\frac{d}{dp} Area $$
derive the area of a circle with respect to the x-axis $$\frac{d}{dx} Area $$

Remember, the derivative just splits the shape into (hopefully) easy-to-measure steps, such as rings of size $$$2 \pi r \ dr$$$. We broke apart our lego set and have pieces scattered on the floor. We still need an integral to glue the parts together and measure the new size. The two commands are a tag team:

Here's how we'd write the integrals to measure the steps we've made:

Formal description Symbol Measures Total Size Of
integrate 2 * pi * r * dr from r=0 to r=r $$\int_{0}^{r} 2 \pi r \ dr$$
integrate [a pizza slice] from [p = min perimeter] to [p = max perimeter] $$ \int_{p=min}^{p=max} (pizza \ slice) \ dp $$
integrate [a board] from [x = min value] to [x = max value] $$ \int_{x=min}^{x=max} board \ dx $$

A few notes:


1) Can you think of another activity which is made simpler by shortcuts and notation, vs. written English?

2) Interested in performance? Let's drive the calculus car, even if you can't build it yet.

Question 1: How would you write the integrals that cover half of a circle?

Circles and spheres

Each should would be similar to:

integrate [size of step] from [start] to [end] with respect to [path variable]

(Answer for the first half and the second half. This links to Wolfram Alpha, an online calculator, and we'll learn to use it later on.)

Question 2: Can you find the complete way to describe our "pizza-slice" approach?

Circles and spheres

The "math command" should be something like this:

integrate [size of step] from [start] to [end] with respect to [path variable]

Remember that each slice is basically a triangle (so what's the area?). The slices move around the perimeter (where does it start and stop?). Have a guess for the command? Here it is, the slice-by-slice description.

Question 3: Can you figure out how to move from volume to surface area?

Circles and spheres

Assume we know the volume of a sphere is 4/3 * pi * r^3. Think about the instructions to separate that volume into a sequence of shells. Which variable are we moving through?

derive [equation] with respect to [path variable]

Have a guess? Great. Here's the command to turn volume into surface area.

Get Free Lessons And Updates

Join 250k monthly readers and Learn Right, Not Rote.

Class Discussion

Guided class discussions are available in the complete edition.