What is the general equation for the arclength of a line?

Mylee Underwood

Mylee Underwood

Answered

2022-07-04

What is the general equation for the arclength of a line?

Answer & Explanation

Perman7z

Perman7z

Expert

2022-07-05Added 13 answers

The general equation of a line is y=mx+b
Recall the formula for arc length is A = a b 1 + ( d y d x ) 2 d x
The derivative of the linear function is y'=m
A = a b 1 + m 2 d x
m is simply a constant, we can use the power rule to integrate.
A = [ 1 + m 2 x ] a b
A = b 1 + m 2 a 1 + m 2
A = ( b a ) 1 + m 2
Now let's verify to see if our formula is correct. Let y=2x+1 and the arc length we wish to find being on the x-interrval [2,6]
A = ( 6 2 ) 1 + 2 2 = 4 5
If we were to use pythagoras, by connecting a horizontal line to a vertical line, we would get the following"
y(2)=5
y(6)=13
y = 13 5 = 8
x = 4
Thus A 2 = 2 y + 2 x = 8 2 + 4 2
A = 80 = 16 5 = 4 5
As obtained using our formula.
fythynwyrk0

fythynwyrk0

Expert

2022-07-06Added 7 answers

For the arc length of a linear function given its slope m and an interval [a,b], using the arc length formula:
S = a b 1 + ( d y d x ) 2 d x
Let y=mx+b
d y d x = m
S = a b 1 + m 2 d x
This may look scary because of all of the variables, but m is technically just a constant: the slope of the line.
The antiderivative is 1 m 2 x, and substituting the limits of integration:
S = 1 m 2 b 1 m 2 a
S = ( b a ) 1 m 2

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