Teaching objectives
How much is the area under a curve worth? Before any formula, this lab starts by counting grid squares by eye. From there it moves to Riemann sums — the rectangles that approximate that area under different criteria — and watches how, as their number grows, two different bounds converge to the same value: the definite integral. With that idea already built, the lab opens up to its applications: signed area, area between curves, and two less obvious uses of the same "slice and add" idea — the length of a curve and the volume of a solid of revolution. Designed for University (Calculus).
What you learn
- Bound the area under a curve by counting grid cells from above and below, and see the estimate improve as the grid gets finer (rooms 1-2).
- Build Riemann sums using the left, right, midpoint, upper and lower criteria, and compare all five at once across several functions (rooms 3-5).
- Watch the upper and lower sums converge to the same value as the number of rectangles grows without bound: that limit is the definite integral (room 6).
- Understand signed area (pieces below the axis subtract), the basic properties of the integral, and why the fundamental theorem of calculus works (rooms 7-9).
- Apply the integral to the area between curves — including curves that cross —, compare numerical methods (Trapezoid, Simpson, Monte Carlo), and reuse the same "slice and add" idea to compute the length of a curve and the volume of a solid of revolution (rooms 10-15).
- Close with an integrative problem that combines a visual estimate, Riemann sums, and the exact integral (room 16).
Key mathematical ideas
- Every Riemann sum approximates the area under a curve with rectangles; the criterion (left, right, midpoint, upper, lower) decides which height each one uses.
- The upper sum and the lower sum trap any reasonable sum between them, and converge to the same value as the number of rectangles tends to infinity: that limit is the definite integral ∫ₐᵇf(x)dx.
- The definite integral measures signed area: pieces below the x-axis subtract instead of adding.
- The fundamental theorem of calculus connects the integral with the derivative: for a strip narrow enough, f behaves almost like a constant (its tangent line), and its area is better and better approximated by a rectangle.
- The area between curves is the integral of the difference between the top and bottom functions; when curves cross, you must split at the crossing points.
- The same partitioning idea generalizes to the length of a curve (approximating it with straight segments) and to the volume of a solid of revolution (approximating it with stacked discs).
- Numerical methods — Trapezoid, Simpson, Monte Carlo — are alternative ways to approximate an integral when there is no simple antiderivative, each with its own order of accuracy.
Room-by-room contents
Room 1 · Area under the curve, by eye
The lab starts without formulas: a curve over a grid and the student marking squares. First they bound the area from above, then from below, and finally give their best estimate. This sets up from the outset the idea that an area can be trapped between two numbers that keep getting closer.
Student tasks
- Mark the squares that cover the area and those that fit entirely beneath the curve.
- Give your own estimate and compare it with the two bounds.
Room 2 · A finer grid
The same curve with a much finer grid. The two bounds narrow and the estimate improves without any change in the method: the precision came from the size of the square. This is the first appearance of the idea of passing to the limit.
Student tasks
- Repeat the count with smaller squares.
- Compare the bounds with those of the previous room and explain why they improve.
Room 3 · Left, right, midpoint
Rectangles replace the squares. A partition of the interval and three ways of choosing each rectangle's height — at the left end, at the right end or at the midpoint — show that the same idea has variants, and that some fall short while others overshoot.
Student tasks
- Compare the three ways of choosing the height on the same partition.
- Decide which one overestimates and which one underestimates for an increasing curve.
Room 4 · Upper and lower sums
Instead of picking an arbitrary point in each subinterval, we take the greatest and the least value of the function on it. The true area is then always trapped between the two sums, whatever the curve does: this is the rigorous version of what was done by eye in room 1.
Student tasks
- Build the upper and the lower sum for the same partition.
- Check that the true area falls between them.
Room 5 · The Riemann explorer
The free-play room of the block: the five selection criteria, a slider for the number of rectangles and several functions to try. It shows at a glance that as n grows all the variants converge to the same number, although some do so much faster than others.
Student tasks
- Increase the number of rectangles and watch the sum settle.
- Compare the speed of convergence of the different criteria.
Room 6 · Convergence and the integral
The upper and lower sums chase each other until they meet at a single number, and that number is the definite integral. Here the ∫ notation is finally introduced and its parts are explained: the elongated sign that stands for a sum, the function and the differential.
Student tasks
- Follow the convergence of the two sums as the partition is refined.
- Relate each part of the notation to what has been done so far.
Room 7 · Signed area
The integral is not always an area: whatever lies below the axis is subtracted. For the sine over a full period the result is zero, and the room ends with the swimming-pool challenge — a function with a constant part and an oscillating part — where the trick is to separate the rectangle from the piece already known to integrate to zero.
Student tasks
- Predict the sign of the integral of a function that crosses the axis.
- Solve the swimming-pool challenge by separating the constant part from the oscillating one.
Room 8 · Properties of the integral
Splitting the interval at an interior point and adding the two pieces gives the same result as integrating in one go. The animation first runs over the whole interval, then over the first piece, and shows the subtraction, which is how this property is actually used in calculations.
Student tasks
- Check additivity by splitting the interval at a point.
- Compute an integral as the difference of two others.
Room 9 · The fundamental theorem
The heart of the lab. If we define the function giving the accumulated area up to x, its derivative turns out to be f itself. The room shows this by zooming in on a very narrow strip: the amount the area grows when we advance a little is, almost exactly, a rectangle of height f(x).
Student tasks
- Estimate how much the area grows when x advances a little.
- Recognize that this growth is the height of the function.
Room 10 · f(x) and its area function
The two graphs side by side: the function and its accumulated area, with a slider that moves along both at once. Where f is positive the area rises, where it is negative the area falls, and where f crosses the axis the area function has a maximum or a minimum.
Student tasks
- Relate the sign of f to the growth of the area function.
- Locate the extrema of the area function from the zeros of f.
Room 11 · Draw your function
Students sketch any curve they like by hand and the lab computes its integral and area function live. It is the room for checking by yourself what the previous two have explained, deliberately drawing curves that rise, fall or cross the axis.
Student tasks
- Draw a free function and observe its area function.
- Deliberately draw a curve whose integral is zero.
Room 12 · Barrow's Rule
The fundamental theorem becomes a computational tool: if an antiderivative is known, the definite integral is the difference of its values at the endpoints. This is where the lab moves from understanding the integral to knowing how to compute it.
Student tasks
- Compute a definite integral by finding an antiderivative and evaluating it.
- Check the result against the numerical estimate.
Room 13 · Area between two curves
The area enclosed between two functions is the integral of their difference. Once the upper and the lower curve are correctly identified, the problem reduces to the usual one.
Student tasks
- Identify which curve is the upper one on the interval.
- Compute the area enclosed between the two.
Room 14 · Curves that cross
When the two curves cross, the upper one stops being the upper one halfway through, so the interval has to be split at the crossing points and the pieces added. This is the classic mistake of the topic, and here you can see why direct subtraction gives a wrong result.
Student tasks
- Find the crossing points of the two curves.
- Split the interval and add the area of each piece.
Room 15 · Trapezoid and Simpson
Two better ways of approximating than rectangles: joining the points with segments or with parabolas. The error bars, on a logarithmic scale, show how much each method improves when the number of pieces is doubled, and there Simpson is seen to win by far more than a little.
Student tasks
- Compare the error of Riemann, trapezoid and Simpson with the same number of pieces.
- Estimate how much each method improves when the pieces are doubled.
Room 16 · Monte Carlo integration
A completely different method: throw random points inside a rectangle and count the proportion that falls under the curve. It is slow and noisy in one dimension, but it opens the door to the idea that there are probabilistic methods for computing integrals.
Student tasks
- Throw points and estimate the area from the proportion that lands below the curve.
- Observe how the estimate improves as the number of points grows.
Room 17 · The length of a curve
The same idea of slicing, applied to another quantity: approximate the curve by a polygonal line and add up the lengths of the segments. As the pieces are refined, the arc-length formula appears.
Student tasks
- Approximate the length with a polygonal line and refine the pieces.
- Recognize where the square root in the formula comes from.
Room 18 · Volume of revolution
The close of the lab, in three dimensions: the curve rotates about the axis and the resulting solid is approximated by discs, each of radius f(x). The scene can be rotated with the mouse, and the volume comes from adding up the discs exactly as the area came from adding up rectangles.
Student tasks
- Rotate the curve and see the solid made of discs.
- Compute the volume by adding the discs and compare it with the formula.
Rooms to project
The most striking ones to show and discuss in class.