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

Question

Evelyn Agard · Accepted Answer

If we wish to find the arc length of #y = mx + b# on #[a, b]#, then #(b - a)sqrt(1 + m^2)# will give the correct arc length.

The general equation of a line is #y = mx + b#. Recall the formula for arc length is #A = int_a^b sqrt(1 + (dy/dx)^2)dx#. The derivative of the linear function is #y' = m#. #A = int_a^b sqrt(1 + m^2)dx# #m# is simply a constant, we can use the power rule to integrate. #A = [sqrt(1+ m^2)x]_a^b# #A = bsqrt(1 + m^2) - asqrt(1 + m^2)# #A = (b - a)sqrt(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)sqrt(1 + 2^2) = 4sqrt(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# #Delta y = 13 - 5 = 8# #Delta x = 4# Thus #A^2 = Delta^2y + Delta^2x = 8^2 +4^2# #A = sqrt(80) = sqrt(16 * 5) = 4sqrt(5)# As obtained using our formula. Hopefully this helps!

Charles Anctil · Answer

#S = (b - a)sqrt(1 + m^2)# For the arc length of a linear function given its slope #m# and an interval #[a, b]#, using the arc length formula: #S = int_a^b sqrt(1 + (color(red)(dy/dx))^2)dx# Let #y = mx + b# #=> color(red)(dy/dx = m)# #S = int_a^b sqrt(1 + m^2) dx# 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 #sqrt(1 - m^2) * x#, and substituting the limits of integration: #S = sqrt(1 - m^2) * b - sqrt(1 - m^2) * a# #S = (b - a)sqrt(1 - m^2)#

Axel Alvarez · Answer

undefined The general equation for the arc length (s) of a line segment in a Cartesian coordinate system between two points (x₁, y₁) and (x₂, y₂) is given by:

$$ s = \sqrt{(x₂ - x₁)^2 + (y₂ - y₁)^2} $$