Sizu Riatto
Calculus · Chapter 18

Slice Adder

When the rate is a curve, cut the time into thin slices, treat each as a rectangle (or a trapezoid), and add. Thinner slices, truer total.

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Opens on Chapter 18’s example: the go-kart’s speed v = t² feet a second, from 0 to 4 seconds, in four slices.

1. From a formula
2. From readings
Enter your numbers, then press Work it out.

How it works

Each slice is a rectangle: width × the rate at its left edge, its right edge or its middle. For a rising curve the left sum is too low and the right too high; the midpoint and the trapezoid (the average of the two edges) are much closer. Double the slices and the error of the trapezoid and the midpoint falls to about a quarter. From readings taken at an even spacing, the trapezoid total is spacing × (all the readings added − half the first − half the last). The exact area, the limit of the slices, is Chapter 19's integral.

From the book, Chapter 18. Four slices under t²: left 14, right 30, midpoint 21, trapezoids 22; the exact area is 21.333 (64/3). With 100 slices the left and right sums are 21.0144 and 21.6544. The roof’s day comes to 21.05 kWh; Jonas’s sprint to 1,085 meters; the storm to about 1.53 inches; Archimedes’ ten slices to 0.385. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.