Let-f-x-x-x-10-and-let-A-be-the-region-enclosed-within-the-following-points-2-7-8-7-2-4-8-4-what-is-the-average-arc-length-of-a-f-x-inside-A-a-R-

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

Let f (x) = x (x - 10)

and let A be the region enclosed within

the following points

(2, 7), (8, 7), (2, 4), (8, 4)

what is the average arc length of a \cdot f (x)

inside A, a \in R^{-}

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

{w h a t}^{'} s y o u r d e f i n i t i o n f o r “ a v e r a g e

a r c {l e n g t h}^{″} ?

Commented by Red1ight last updated on 22/Apr/24

Sum of lengths for all possible arcs

that can be drawn

within that region

devided by the range of a

avg = \frac{Σ L_{i}}{a_{m a x} - a_{m i n}}

where L_{i} is a length of an arc when a = a_{i}

where it is also within the region

where a_{m i n} ⩽ a_{i} ⩽ a_{m a x}

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

i a s k e d t h i s b e c a u s e y o u g a v e a \in R^{-},

w h i c h m e a n s a_{m i n} = - \infty, a_{m a x} = 0 .

Answered by Frix last updated on 22/Apr/24

y = a x (x - 10)

Arc length α ⩽ x ⩽ β : \underset{α}{\int^{β}} \sqrt{1 + {(y^{'})}^{2}} d x

1 . - \frac{7}{16} ⩽ a ⩽ - \frac{7}{25}

2 interections with y = 7 : x = 5 \pm \sqrt{\frac{25 a + 7}{a}}

\frac{400}{63} \underset{- \frac{7}{16}}{\int^{- \frac{7}{25}}} (2 \underset{2}{\int^{5 - \sqrt{\frac{25 a + 7}{a}}}} \sqrt{1 + {(y^{'})}^{2}} d x) d a \approx 3.05122

2 . - \frac{7}{25} ⩽ a ⩽ - \frac{1}{4}

\frac{100}{3} \underset{- \frac{7}{25}}{\int^{- \frac{1}{4}}} (2 \underset{2}{\int^{5}} \sqrt{1 + {(y^{'})}^{2}} d x) d a \approx 7.98232

3 . - \frac{1}{4} ⩽ a ⩽ - \frac{4}{25}

2 intersections with y = 4 : x = 5 \pm \sqrt{\frac{25 a + 4}{a}}

\frac{100}{9} \underset{- \frac{1}{4}}{\int^{- \frac{4}{25}}} (2 \underset{5 - \sqrt{\frac{25 a + 4}{a}}}{\int^{5}} \sqrt{1 + {(y^{'})}^{2}} d x) d a \approx 5.06862

Sum of these \approx 16.1022

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