\frac{a ^{3}}{3}

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1598 – 1647Milan, Italy

Bonaventura Cavalieri

The area under $x^{n}$ follows a simple pattern.

A short story

He was a member of the Order of Jesuates (an Italian religious order, distinct from the Jesuits) and one of Galileo's students. In 1635 he published a book where he showed that the area under $y = x^{n}$ from $0$ to $a$ is $a^{n + 1} / (n + 1)$ , fifty years before Newton and Leibniz invented calculus. His method of indivisibles sliced shapes into infinitely thin slivers and added them up. The same idea is the heart of integration today.

In their own words

If two solids have equal cross-sections at every height, they have equal volumes.

Paraphrased: Cavalieri's principle, from Geometria indivisibilibus continuorum, 1635.

The lab their idea turned into

Basin Lab

Measure the space scooped out by any parabola of the form $x^{2} + b x + c$ .

Open Basin Lab →