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Proof without words — page 4

Arguments that are complete once they have been looked at properly. Not illustrations of proofs — the proofs themselves.
A circular sector of area 1.10 and a hyperbolic sector of area 0.80. On the left, the unit circle with the sector from (1, 0) to (cos 2.2, sin 2.2), of area 1.100. On the right, the hyperbola x² − y² = 1 with the sector from (1, 0) to (cosh 1.6, sinh 1.6), of area 0.800. In both, the parameter is twice the shaded area. Analysis

The angle that is really an area

On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.

 The sum of the first 6 squares, as a staircase over a curve. Bars of height k^2 for k from 1 to 6, totalling 91, drawn over the curve y = x^2, whose area up to 6 is 72.00. The slivers between staircase and curve hold 19.00, close to half the last bar. Geometry

Sums of powers, read off a staircase

Add the first n squares, or cubes, or seventh powers, and the answer is always a polynomial in n. Its first term is the area under a curve, its second is half of the last step, and every term after that is a correction for the corners of a staircase — which is where the Bernoulli numbers come from, and why they eventually grow without bound.

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