area of one loop of a cycloid?This area finding technique is not so important, so since time is short, we will not talk about it.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 area under one arc of the cycloid [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] is easily done by evaluating the line integral over the upper arc completed by the line segment along the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn axis connecting the endpoints, using Green's theorem applied to the special vector field whose Green's theorem double integral integrand is 1, so produces the area of the region enclosed by the loop.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Not also that this vector field is vertical along the x-axis where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 so
the line integral along the x-axis does not contribute to the left hand side of Green's theorem.
Because there is no simple way to express y as a function of x for the cycloid, we cannot apply the calc2 approach of finding the area between two curves and can only evaluate it using Green's theorem! [We can express x as a function of y over half a cycloid loop, see below but it is very complicated.]
This example shows the reverse utility of Green's theorem, that in some cases it is easier to do the line integral instead of the double integral, in contrast with the simpler examples where the double integral is usually easier to do than the line integral.
The cycloid in particular cannot be directly represented as the graph of a simple function (see below), so the direct area integration approach is complicated, while the Green's theorem approach leads to a trivial integral that can be done by hand, even in our heads.line integral approachThe function graph is the upper boundary of the region, and so must be traversed in the reverse-x direction, so one must reverse the sign of the integral to 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 x-axis boundary of the region contributes 0 to the total line integral since the integrand is zero: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So the total area under one loop is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEiM0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSNtaUdGJDYmUScmIzk2MDtGJy8lJ2l0YWxpY0dGMUYvRjJGL0Yy. This means that the average value of y over the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjwtRjk2LlEjLi5GJ0YyRjxGPkZARkJGREZGRkhGSi9GTVEsMC4yMjIyMjIyZW1GJy9GUFEmMC4wZW1GJy1GUjYlUSIyRidGMkY8LUYjNiYtRiw2I1EhRictRiw2JlElJnBpO0YnRi9GMkY1RjJGPEZbb0YyRjw= one loop 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direct function graph approach leads to an ugly 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One can only solve the second relation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, and then express LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn for half an arc of a cycloid period, and this would be a nightmare to integrate in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn direction first and then in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn direction, so a double integral to find the area under one cycloid arc would not be very useful to find the exact result by hand (although Maple can still do the integral exactly).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To find the area under this curve as a double integral: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JSFHsame deal for the ellipse example in the textbook (line integral trivial)The ellipse is easily parametrized as a deformed circle.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 total area is twice the integral of the top function between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEqJnVtaW51czA7RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZFLUkjbW5HRiQ2JVEiM0YnRi9GMkYvRjI= and 3 (top plus bottom halves):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 Maple, this is no big deal but for us the antiderivative requires advanced techniques. However, one can also reintepret the integral as a multiple of the area under a semicircle to get the same result. The equivalent line integral is much simpler and can be done by hand.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JVEjUjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSomY29sb25lcTtGJ0YvRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi1GLDYlUSJ0RidGL0YyLUY2Ni5RKCZzcmFycjtGJ0YvRjJGOUY8Rj5GQEZCRkRGRi9GSVEmMC4wZW1GJy9GTEZULUkobWZlbmNlZEdGJDYmLUYjNiYtRiw2I1EhRictRiM2LC1JI21uR0YkNiRRIjNGJy9GM1Enbm9ybWFsRictRjY2LVEifkYnRl5vRjlGPEY+RkBGQkZERkZGU0ZVLUYsNiVRJGNvc0YnL0YwRjtGXm8tRlc2JC1GIzYkRk1GXm9GXm8tRjY2LVEiLEYnRl5vRjkvRj1GMUY+RkBGQkZERkZGUy9GTFEsMC4zMzMzMzMzZW1GJy1GW282JFEiMkYnRl5vRmBvLUYsNiVRJHNpbkYnRmZvRl5vRmdvRl5vRmVuRl5vRl5vLyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJy1GNjYtUSI6RidGXm9GOUY8Rj5GQEZCRkRGRkZIRktGYG9GK0Znby1GNjYtUSI7RidGXm9GOUZecEY+RkBGQkZERkZGU0ZLRmBvRistRjY2LVEiJ0YnRl5vRjlGPEY+RkBGQkZERkYvRklRLDAuMTExMTExMWVtRidGVUZnb0ZgcS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0ZULyUmZGVwdGhHRl1yLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUYsNiVRJEYxdEYnRi9GMi1GNjYtRjhGXm9GOUY8Rj5GQEZCRkRGRkZIRktGTS1GNjYtRlJGXm9GOUY8Rj5GQEZCRkRGRkZTRlUtSSZtZnJhY0dGJDYoLUZbbzYkUSIxRidGXm8tRiM2JEZhcEZeby8lLmxpbmV0aGlja25lc3NHRmFzLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmhzLyUpYmV2ZWxsZWRHRjtGYG8tRlc2Ji1GIzYtLUY2Ni1RKiZ1bWludXMwO0YnRl5vRjlGPEY+RkBGQkZERkYvRklRLDAuMjIyMjIyMmVtRicvRkxGZXRGYXBGYG9GZHBGZ29GW3BGam5GYG9GY29GZ29GXm9GXm9GZ3BGanBGXXFGYG9GZXJGZ29GXm8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2JVEkRjF0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidEYnRi9GMi9GM1Enbm9ybWFsRidGPS1JI21vR0YkNi1RKSZtaWRkb3Q7RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZULUYsNiVRI1IxRidGL0YyLUZANi1RIidGJ0Y9RkNGRkZIRkpGTEZORlAvRlNRLDAuMTExMTExMWVtRicvRlZRJjAuMGVtRidGNS1GQDYtUSI7RidGPUZDL0ZHRjFGSEZKRkxGTkZQL0ZTRmpuL0ZWUSwwLjI3Nzc3NzhlbUYnLUZANi1RIn5GJ0Y9RkNGRkZIRkpGTEZORlBGX29GaW5GYm8tRiw2JVEpc2ltcGxpZnlGJ0YvRjItRjY2JC1GIzYkLUYsNiVRIiVGJ0YvRjJGPUY9RltvLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRmpuLyUmZGVwdGhHRmRwLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUkobXN1YnN1cEdGJDYnLUZANi5RKyZJbnRlZ3JhbDtGJy8lK2V4ZWN1dGFibGVHRkVGPUZDRkYvRklGMUZKL0ZNRjFGTkZQRl9vRmluLUYjNiQtSSNtbkdGJDYkUSIwRidGPUY9LUYjNiYtRmlxNiRRIjJGJ0Y9RmJvLUYsNiVRJyYjOTYwO0YnL0YwRkVGPUY9LyUxc3VwZXJzY3JpcHRzaGlmdEdGW3IvJS9zdWJzY3JpcHRzaGlmdEdGW3JGXHBGYm8tRkA2LVEwJkRpZmZlcmVudGlhbEQ7RidGPS9GRFEmdW5zZXRGJy9GR0Zdcy9GSUZdcy9GS0Zdcy9GTUZdcy9GT0Zdcy9GUUZdc0Zfb0ZpbkY6Rl9wLUYsNiNRIUYnRj0=In fact the upper half ellipse with semiaxis 3 encloses the same area 3LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiw2JVElJnBpO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GNlEnbm9ybWFsRic= as the cycloid with height 2. Compare their graphs.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 the average value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn for the (blue) ellipse is slightly higher than the (red) cycloid.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 the other hand we are not comparing the same intervals. If we stretch out the ellipse to 2LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiw2JVElJnBpO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GNlEnbm9ybWFsRic= (with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GIzYmLUYsNiNRIUYnLUYsNiZRJSZwaTtGJ0YvRjJGNUYyRjxGU0YyRjw= instead of 3, so the areas are no longer equal) then we get the same average value (we also extended the horizontal semi-axis!), which is still a bit higher than the cycloid.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The ends explain why the ellipse is higher than the cycloid, since there one sees the biggest difference.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It helps to know the formula in general for the area of half an ellipse with semiaxes a and b letting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GIzYmLUYsNiNRIUYnLUYsNiZRJSZwaTtGJ0YvRjJGNUYyRjxGU0YyRjw=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JlEsQXJlYUVsbGlwc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tRjk2LlEifkYnRjJGPEY+RkBGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQRlUtRiw2JlEnJiM5NjA7RidGL0YyRjUtRjk2L0ZTRi9GMkY1Rj5GQEZCRkRGRkZIRkpGVEZWLUYsNiZRImFGJ0YvRjJGNUZaLUYsNiZRImJGJ0YvRjJGNS1GOTYvUSI7RidGL0YyRjVGPi9GQUYxRkJGREZGRkhGSkZURk8tRiw2JlEwQXJlYUhhbGZFbGxpcHNlRidGL0YyRjVGOC1JJm1mcmFjR0YkNigtSSNtbkdGJDYlUSIxRidGMkY8LUYjNiYtRmdvNiVRIjJGJ0YyRjxGL0YyRjUvJS5saW5ldGhpY2tuZXNzR0Zpby8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZjcC8lKWJldmVsbGVkR0Y0RlFGV0ZaRmZuRlpGaW5GUUY4RlFGY29GUS1JJW1zdXBHRiQ2JS1GLDYmUSUmcGk7RidGL0YyRjVGam8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRlFGaW5GMkY8LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEseWF2Z0VsbGlwc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSZtZnJhY0dGJDYoLUYjNiktRlA2KC1JI21uR0YkNiRRIjFGJ0Y5LUYjNiUtRlc2JFEiMkYnRjkvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGOS8lLmxpbmV0aGlja25lc3NHRlkvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGPS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORmlvLUklbXN1cEdGJDYlLUYsNiVRJSZwaTtGJ0YvRjJGWi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGZW8tRiw2JVEiYkYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOS1GIzYpRmZuLUYjNiUtRiw2I1EhRidGXnBGOUZdcUYvLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGMUYyRlxvRl5vRmFvRmNvLUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGaG9GTS1GLDYlUSVzdWJzRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNilGZHBGNUZmbi1GNjYtUSIsRidGOUY7RmhxRkBGQkZERkZGSEZoby9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSIlRidGL0YyRmdwRjlGOUZlcUZlby1GLDYlUSZldmFsZkYnRi9GMi1GXXI2JC1GIzYlRmZyRmdwRjlGOUZncEY5The area under a cycloid loop with only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn scaled up to the same vertical height 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Math people are curious, so tend to go off on tangents like this exploring mathematical relationships.