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M04.5 · Integrals as accumulated area

A free, illustrated module of 4 lessons and one interactive in M04 Calculus I: Single Variable: Distance from speed: adding up thin slices; Riemann sums and the definite integral; Signed area, properties and the average value; The trapezoid rule and integrals of sampled data.

M04.5 L1 · Core 4 lessons · about 100 min Live · in review

Part of M04 Calculus I: Single Variable in the Mathematics track.

A speed in metres per second, a power in watts or a rate of examples per second tells you how fast something piles up; the total is the area under its graph. These four lessons build that area from thin rectangles, trap a distance between a left and a right sum, let the strips shrink to define the definite integral (and prove that the area under x squared from 0 to 1 is exactly 1/3), count area below the axis as negative and read off averages and probabilities, and finish with the trapezoid rule, which integrates a table of samples such as an ROC curve.

What you need first. Sigma notation (M01.7.2), proof by induction (M01.6.4) and limits as n grows (M04.1.3), and the derivative as a rate of change (M04.2), with the running trip’s velocity found from its position (M04.3.1). The mean value theorem (M04.4.1) appears once, in a collapsed proof.

Lessons

1. Distance from speed: adding up thin slices M04.5.1 25 min

   At a steady speed, distance is speed times time, the area of a rectangle under the speed graph. When the speed changes, cut time into thin slices, treat each as steady and add the rectangles. The left and right sums trap the true distance, and the gap between them shrinks with the slices.

2. Riemann sums and the definite integral M04.5.2 25 min

   Cut an interval into equal strips, take a height in each, and add height times width: a Riemann sum. For x squared on [0, 1] the right sums have a closed form that tends to exactly 1/3, and that limit is the definite integral. Compute the sums by hand, by a loop and with NumPy.

3. Signed area, properties and the average value M04.5.3 25 min

   Area below the axis counts as negative, so an integral is a net amount: a velocity integrates to net distance, not distance travelled. Derive linearity, splitting, reversing and comparison from the sums, then use them for the average value of a function and a first probability.

4. The trapezoid rule and integrals of sampled data M04.5.4 25 min

   Join neighbouring samples with straight lines and the rectangles become trapezoids: the average of the left and right sums, and far more accurate. The rule needs only samples, so it integrates tables of data, from a cart’s speedometer readings to the points of an ROC curve.

The interactive

M04.5-X1 · Riemann refiner. Choose a function and a rule (left, right, midpoint or trapezoid) and slide the number of strips from 1 to 64. The strips redraw under the curve, and the read-out gives the sum, the exact area, the error and how much the error shrinks each time the strips double. It lives in Riemann sums and the definite integral.