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Question Number 206645 by Red1ight last updated on 21/Apr/24

$$\mathrm{Let}f\left(x\right)=x(x-10)$$$$\mathrm{and}\mathrm{let}\mathrm{A}\mathrm{be}\mathrm{the}\mathrm{region}\mathrm{enclosed}\mathrm{within}$$$$\mathrm{the}\mathrm{following}\mathrm{points}$$$$(2,7),(8,7),(2,4),(8,4)$$$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{average}\mathrm{arc}\mathrm{length}\mathrm{of}\mathrm{a}\cdot f\left(x\right)$$$$\mathrm{inside}\mathrm{A},a\in {\mathbb{R}}^{-}$$

Commented by mr W last updated on 21/Apr/24

$${what}^{\prime }syourdefinitionfor‘‘average$$$$arc{length}^{″}?$$

Commented by Red1ight last updated on 22/Apr/24

$$\mathrm{Sum}\mathrm{of}\mathrm{lengths}\mathrm{for}\mathrm{all}\mathrm{possible}\mathrm{arcs}$$$$\mathrm{that}\mathrm{can}\mathrm{be}\mathrm{drawn}$$$$\mathrm{within}\mathrm{that}\mathrm{region}$$$$\mathrm{devided}\mathrm{by}\mathrm{the}\mathrm{range}\mathrm{of}\mathrm{a}$$$$\mathrm{avg}=\frac{\mathrm{\Sigma }{L}_i}{a_{max}-a_{min}}$$$$\mathrm{where}{\mathrm{L}}_i\mathrm{is}\mathrm{a}\mathrm{length}\mathrm{of}\mathrm{an}\mathrm{arc}\mathrm{when}\mathrm{a}={\mathrm{a}}_{\mathrm{i}}$$$$\mathrm{where}\mathrm{it}\mathrm{is}\mathrm{also}\mathrm{within}\mathrm{the}\mathrm{region}$$$$\mathrm{where}{a}_{min}⩽{\mathrm{a}}_{\mathrm{i}}⩽a_{max}$$

Commented by Red1ight last updated on 22/Apr/24

$$$$for example the green parabola represents a=-0.1 which from yhe plot is it not within the region while the red and blue (a=-0.2,a=-0.3) parabola are within. so their arc length is included in the summation

Commented by Red1ight last updated on 22/Apr/24

Commented by mr W last updated on 22/Apr/24

$$iaskedthisbecauseyougavea\in R^{-},$$$$whichmeans{a}_{min}=-\mathrm{\infty },a_{max}=0.$$

Answered by Frix last updated on 22/Apr/24

$$y=ax(x-10)$$$$\mathrm{Arc}\mathrm{length}\alpha ⩽x⩽\beta :\underset{\alpha }{\stackrel{\beta }{\int }}\sqrt{1+{\left(y^{\prime }\right)}^2}dx$$$$1.-\frac{7}{16}⩽a⩽-\frac{7}{25}$$$$2\mathrm{interections}\mathrm{with}y=7:x=5\pm \sqrt{\frac{25a+7}{a}}$$$$\frac{400}{63}\underset{-\frac{7}{16}}{\stackrel{-\frac{7}{25}}{\int }}\left(2\underset{2}{\stackrel{5-\sqrt{\frac{25a+7}{a}}}{\int }}\sqrt{1+{\left(y^{\prime }\right)}^2}dx\right)da\approx 3.05122$$$$2.-\frac{7}{25}⩽a⩽-\frac{1}{4}$$$$\frac{100}{3}\underset{-\frac{7}{25}}{\stackrel{-\frac{1}{4}}{\int }}\left(2\underset{2}{\stackrel{5}{\int }}\sqrt{1+{\left(y^{\prime }\right)}^2}dx\right)da\approx 7.98232$$$$3.-\frac{1}{4}⩽a⩽-\frac{4}{25}$$$$2\mathrm{intersections}\mathrm{with}y=4:x=5\pm \sqrt{\frac{25a+4}{a}}$$$$\frac{100}{9}\underset{-\frac{1}{4}}{\stackrel{-\frac{4}{25}}{\int }}\left(2\underset{5-\sqrt{\frac{25a+4}{a}}}{\stackrel{5}{\int }}\sqrt{1+{\left(y^{\prime }\right)}^2}dx\right)da\approx 5.06862$$$$\mathrm{Sum}\mathrm{of}\mathrm{these}\approx 16.1022$$

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