The Area Under the Curve
Like Duolingo, but for The Area Under the Curve. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
For the part of you with thirty open tabs that never became anything.
Free during early access · No credit card · iPhone & Android
Key ideas in The Area Under the Curve
- Thinner rectangles have less 'overhang' or empty space sticking out past the curve
- Smaller pieces follow the slope of the curve more closely than wide ones
- The process starts with a rough approximation using simple shapes
- The exact area is reached only when the pieces become infinitely thin
- Why increasing the number of rectangles improves the area estimate
- The conceptual process of reaching an exact area through subdivision
- At an infinitesimal scale, a tiny segment of a curve looks like a straight line
- The total area is the sum of these tiny, straight-edged slices
- The logic of using straight edges to represent a curve
- Recognizing when continuous summation is required over simple addition
- Distinguishing between discrete addition and continuous summation
- Break the whole into tiny, thin vertical slices.
- Treat each tiny slice as if it were a flat, fixed value for a split second.
- Add all those infinite tiny slices together to get the true total.
- The conceptual process of building a total from changing values
- A speedometer only shows the rate at one specific moment in time
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives The Area Under the Curve the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
Why does a curve look almost like a flat line if you zoom in on a tiny, small piece of it?
Get it right to open this lesson and 29 more in the app.
Where The Area Under the Curve takes you
Move beyond basic formulas to see how integration measures the accumulation of change in the real world. Master the art of slicing complex shapes into infinite slivers to solve problems in physics, economics, and design.
- 1
Measure the Unmeasurable
- The Power of Infinite Slivers
- Summing Up Continuous Change
- Finding Total Value from a Rate
- The Link Between Slopes and Areas
- Handling Negative Space and Net Change
- 2
Master the Toolkit
- Reversing the Chain Rule
- The Art of Substitution
- Breaking Down Complex Products
- Dealing with Infinite Boundaries
- 3
Apply Your Perspective
- Calculating Work and Energy
- Finding the Center of a Shape
- Predicting Growth and Decay
- Measuring the Length of a Curve
- Volumes of Spinning Objects
- Probability as an Area
3 sections · 15 units · 30 levels. Built to play, not to enroll.
You pick the voice
The Area Under the Curve is taught in the Explain Like I'm 5 style: no big words. promise.. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start The Area Under the Curve today.
Download Tomo, search The Area Under the Curve, and play your first lesson in under a minute.