arclengthCalculus is about zooming into a function till it looks linear, defining linear quantities, then zooming out.The tangent line is obtained this way when it exists. Zoom in at a point on a graph of a function until the curve looks straight; its slope is the value of the derivative, and the line with this slope through the point is the tangent line.
Click on plot below to make it active, then right click to change Manipulator to "Zoom in". Put the cross hairs on the point on the curve where you want to zoom in, then either left click successively or use your mouse scroll to zoom in. Keep the cross hairs on the curve or you will zoom in on white space. You can switch to zoom out to back out or re-execute the plot command to restart at the original plot window.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Here is a zoom plot of the point at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRJTAuNjBGJ0YyRjxGMkY8 on this graph.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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoJiM5MTY7eUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn along the tangent line to a graph LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTkYrLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjJGPEYyRjxGMkY8 are equal to the slope 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times the increment 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 so the increment of arclength LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoJiM5MTY7c0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn along the hypotenuse by the Pythagorean theorem has length
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
so this is the approximate increment that we have to pass to the limit turning the sum over the finite subintervals into a definite integral which then defines what we mean by arclength.
s = 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As you might guess, for the square root of the sum of one and the square of the derivative of a function to be simple enough to have an antiderivative easily expressible in terms of elementary functions, the function must be very special. Indeed, for toy problems to do by hand there are only a few variations of a few simple cases which allow a hand solution without technology. The 3/2 power function is one of them and integrals which can be reduced to it by a simple LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn-substitution, because they make the input to the square root a perfect square, eliminating the root obstacle to integration.numerical approximation visualized (limit of sequences, our next topic!)The Riemann sum definition of an integral, whether it represent an area or a length, is a limit of a sequence of approximation sums of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn terms as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn goes to infinity. Truncating this sequence at a large enough value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn gives a numerical approximation to the integral.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMEYnRjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZaLUYsNiVRImJGJ0YvRjJGNS1GUDYkUSIxRidGOUZTLUYsNiVRImZGJ0YvRjJGNS1GLDYlUSJ4RidGL0YyLUY2Ni1RKCZzcmFycjtGJ0Y5RjtGPkZARkJGREZGRkhGWUZlbi1JJW1zdXBHRiQ2JUZfby1GIzYlLUZQNiRRIjJGJ0Y5Ri9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHRlItRiw2I1EhRicvJStleGVjdXRhYmxlR0Y9Rjk=First we create a plot of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbW5HRiQ2JVEiMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSNtaUdGJDYmUSJORicvJSdpdGFsaWNHUSV0cnVlRidGMi9GNlEnaXRhbGljRidGPkYyRkEvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNQ== secant lines from a division of the graph interval into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbW5HRiQ2JVEiMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSNtaUdGJDYmUSJORicvJSdpdGFsaWNHUSV0cnVlRidGMi9GNlEnaXRhbGljRidGPkYyRkEvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNQ== divisions of width 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. Here is the case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8 of two divisions.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Now we create an animation in which we keep doubling the number of divisions by dividing each subinterval in half, but this moves too quickly to see much so in the next attempt we slow it down by dividing into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn subintervals and increasing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn from 1 to 100, ending at a subinterval of width 0.01.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 slower, then we zoom in with the cursor Manipulator to see that the secant lines eventually pull away from the curve at a small enough viewing window.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRImtGJ0YvRjItRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRImFGJ0YvRjItRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZpbi1JJm1mcmFjR0YkNigtSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJiRidGL0YyLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkhGaG5Gam5GWEY5RjktRiM2JC1GLDYlUSJORidGL0YyRjkvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNwLyUpYmV2ZWxsZWRHRj0tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZVRldGTy1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0ZWLyUmZGVwdGhHRmNxLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZfcTYmRmFxRmRxRmZxL0ZpcVElYXV0b0YnLUYsNiVRKGRpc3BsYXlGJ0YvRjItRl9vNiQtRiM2Ki1GLDYlUSRzZXFGJ0YvRjItRl9vNiQtRiM2LEZfci1GX282JC1GIzYqLUYsNiVRJXBsb3RGJ0YvRjItRl9vNiQtRiM2Ny1GLDYlUSJmRidGL0YyLUZfbzYkLUYjNiQtRiw2JVEieEYnRi9GMkY5RjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZVL0ZOUSwwLjMzMzMzMzNlbUYnRl90LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNRlgtRjY2LVEjLi5GJ0Y5RjtGPkZARkJGREZGRkhGaG5GV0Zjb0ZidC1GLDYlUSZjb2xvckYnRi9GMkZodC1GLDYlUSZibGFja0YnRi9GMkZidC1GLDYlUShzY2FsaW5nRidGL0YyRmh0LUYsNiVRLGNvbnN0cmFpbmVkRidGL0YyRmJ0LUYsNiVRKmdyaWRsaW5lc0YnRi9GMkZodC1GLDYlRjFGL0YyRjlGOUZidEZecUZbckZhcy1GX282JC1GIzYuLUZfbzYmLUYjNiVGZnItRl9vNiQtRiM2Ki1GX282Ji1GIzYoRistRl9vNiQtRiM2JC1GLDYlUSJpRidGL0YyRjlGOUZidEZocy1GX282JC1GIzYlRitGX3dGOUY5RjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZidEZjd0ZodC1JI21uR0YkNiRRIjBGJ0Y5Rlt1RltwRjlGOUY5RjlGandGXXhGYnRGX3RGaHRGWEZbdUZjb0ZidEZedUZodC1GLDYlUSRyZWRGJ0YvRjJGOUY5RjlGOUZidEZbcEZodC1GYXg2JEZgcEY5Rlt1LUZheDYkUSMxMEYnRjkvJStleGVjdXRhYmxlR0Y9RjlGOUZidC1GLDYlUStpbnNlcXVlbmNlRidGL0YyRmh0Rl12Rlx5RjlGOUZceUY5The sum of the secant line segment lengths in these piecewise linear approximations to the graph of the function for a given LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn is easily evaluated in Maple, using the distance formula for the distance between the successive endpoints 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEifkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTkYyRjw=th interval. We use the doubling sequence to more quickly get to a good approximation.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If we increase the number of divisions by consecutive integers the "convergence" of this sequence is much slower to see, because we need to get up to at about 100 divisions before we see the 4 decimal place approximation stabilize. Here are the first 11, starting at the obvious length LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbXNxcnRHRiQ2Iy1GIzYlLUkjbW5HRiQ2JVEiMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjRGNy1JI21vR0YkNi5RKSZhcHByb3g7RidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GMTYlUSYxLjQxNEYnRjRGN0Y0Rjc= of a single division.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 the integrals: some examplesFirst look at three power functions connecting the origin to the point 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graph of the first function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIj1GJ0YyRkAvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRj1GMkZA is just a straight line between the endpoints with an obvious length of 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we need technology to integrate the arclength integral, but an inverse hyperbolic function comes out of the antiderivative, which can be re-expressed in terms of the natural log 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we go directly to the antiderivative, Maple bypasses the hyperbolic function.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What rule of logs makes these two expressions for the same number equivalent?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For 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 this can easily be done by a simple LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn-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Now consider a simple negative power function. For 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, 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which we need technology to integrate the arclength integral, but the result is very complicated and involves complex number expressions, so much so that the numerical value has a small imaginary part from truncation error at the last significant digit. Here because of the division by zero at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8,we shift to over the interval to 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ignore this!
Stewart Calculus 8e example 8.1.4 on steriods, other exercises that allow exact integration "by hand"When the derivative is a difference of power functions which are reciprocally related with very special coefficients, the result can be represented as the square of the corresponding sum.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 one value of the power makes this a perfect square case.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInhGJ0YvRjItRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRImFGJ0YvRjItRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZVRlctSShtZmVuY2VkR0YkNiQtRiM2KC1JJW1zdXBHRiQ2JUZPLUYjNiUtSSNtbkdGJDYkUSIyRidGOUYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GXXAtRiw2JVEjbG5GJy9GMEY9RjktRmluNiQtRiM2JS1GXm82JUZPLUYjNiUtRiw2JVEicEYnRi9GMkYvRjJGZm8vJStleGVjdXRhYmxlR0Y9RjlGOUZecUY5RjlGXnFGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ4RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSC9GS0ZPL0ZOUSwwLjI3Nzc3NzhlbUYnLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGaG5GTS1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GZm9GK0Y1LUklbXN1cEdGJDYlRlAtRiM2JS1GX282JFEiMkYnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZaLUYsNiVRJ2V4cGFuZEYnRi9GMi1GUTYkLUYjNiUtRiw2JVEiJUYnRi9GMkZYRjlGOUZYRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JVEnZXhwYW5kRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtSSVtc3VwR0YkNiUtRjY2JC1GIzYrLUkjbW5HRiQ2JFEiMkYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0ZFLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZNLyUpc3RyZXRjaHlHRk0vJSpzeW1tZXRyaWNHRk0vJShsYXJnZW9wR0ZNLyUubW92YWJsZWxpbWl0c0dGTS8lJ2FjY2VudEdGTS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmZuLUYsNiVRImFGJ0YvRjJGRy1GLDYlUSJ4RidGL0YyLUZINi1RIitGJ0ZFRktGTkZQRlJGVEZWRlgvRmVuUSwwLjIyMjIyMjJlbUYnL0ZobkZjby1JJm1mcmFjR0YkNigtRiM2J0ZpbkZHLUYsNiVRInBGJ0YvRjJGL0YyLUYjNiVGXG9GL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZkcC8lKWJldmVsbGVkR0ZNLyUrZXhlY3V0YWJsZUdGTUZFRkUtRiM2JUZBRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGaXBGRUZFLUZINi1RIj1GJ0ZFRktGTkZQRlJGVEZWRlgvRmVuUSwwLjI3Nzc3NzhlbUYnL0ZobkZkcS1GQjYkRmFwRkVGX28tRkI2JFEiNEYnRkUtRkg2LVExJkludmlzaWJsZVRpbWVzO0YnRkVGS0ZORlBGUkZURlZGWEZaRmduLUY7NiVGaW5GQUZdcUZbci1GOzYlRlxvRkFGXXEtRkg2LVEoJm1pbnVzO0YnRkVGS0ZORlBGUkZURlZGWEZib0Zkb0ZocUZbckZeckZbckZqb0Zfby1GZm82KC1GIzYqRl5yRltyLUY7NiVGam9GQUZdcS8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJ0ZpcC8lKXJlYWRvbmx5R0YxLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGRS1GIzYoRmByRltzRmlwRl5zRmBzRkVGX3BGYnBGZXBGZ3AtRkg2LVEiO0YnRkVGSy9GT0YxRlBGUkZURlZGWEZaRmVxLUYsNiVRKXNpbXBsaWZ5RidGL0YyLUY2NiQtRiM2KS1GLDYlUSRsaHNGJ0YvRjItRjY2JC1GIzYlLUYsNiVRIiVGJ0YvRjJGaXBGRUZFRmJyLUYsNiVRJHJoc0YnRi9GMkZjdEZpcEZFRkVGZXNGam9GYHEtRiw2JVEmc29sdmVGJ0YvRjItRjY2JC1GIzYnRmd0LUZINi1RIixGJ0ZFRktGaHNGUEZSRlRGVkZYRlovRmhuUSwwLjMzMzMzMzNlbUYnRmpvRmlwRkVGRUZpcEZFSuppose we restart with this ratio of the coefficients:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInhGJ0YvRjItRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRImFGJ0YvRjItRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZVRlctSSVtc3VwR0YkNiVGTy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmhvLUkmbWZyYWNHRiQ2KC1GXm82JFEiMUYnRjktRiM2Jy1GXm82JFEiOEYnRjlGZW5GWEYvRjIvJS5saW5ldGhpY2tuZXNzR0ZfcC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZpcC8lKWJldmVsbGVkR0Y9RmVuLUYsNiVRI2xuRicvRjBGPUY5LUkobWZlbmNlZEdGJDYkLUYjNiVGTy8lK2V4ZWN1dGFibGVHRj1GOUY5LUYsNiNRIUYnRmdxRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ4RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSC9GS0ZPL0ZOUSwwLjI3Nzc3NzhlbUYnLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGaG5GTS1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GZm9GK0Y1LUklbXN1cEdGJDYlRlAtRiM2JS1GX282JFEiMkYnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZaLUYsNiVRJ2V4cGFuZEYnRi9GMi1GUTYkLUYjNiUtRiw2JVEiJUYnRi9GMkZYRjlGOUZaLUYsNiVRKXNpbXBsaWZ5RidGL0YyLUZRNiQtRiM2Jy1GaW82JS1GUTYkLUYjNi5GXXAtRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSEZobkZNLUYsNiVRImFGJ0YvRjJGanFGVUZiby1JJm1mcmFjR0YkNigtRiM2KEZeby8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJ0ZYLyUpcmVhZG9ubHlHRjEvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Y5LUYjNiwtRl9vNiRRIjhGJ0Y5RmpxRl1yRmpxRlVGZXJGWEZockZqckY5LyUubGluZXRoaWNrbmVzc0dGYW8vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZnMvJSliZXZlbGxlZEdGPUZlckZYRmhyRmpyRjlGOUZdcEZgcC1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIRmVvRmdvRmpwRlhGOUY5RlhGOQ==What has to be true about the coefficients?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Now we look at some other positive negative power combinations that result in perfect squares that can be used for doable calculus problems.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 how can we quantify these cases to generalize them to a one parameter family?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As long as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVy8lK2V4ZWN1dGFibGVHRj1GOQ== (where the ln is needed) we get the sign reversed function for the arclength antiderivative.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 tells us where the zero of arclength 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This is the condition on the two coefficients in terms of the power chosen.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 is a perfect square case with trig functions (these only differ by a sign when you use the ln rule for exponents! and the sign squares out)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This one is a perfect square but a rational function once you take the square root.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Some hyperbolic/exponential similar 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