The given curve is y = 5x -4, -1 ≤ x ≤ 3.
The length of the arc is computed from the formula
[tex]S= \int_{-1}^{3} \,\sqrt{1+ (\frac{dy}{dx} )^{2}} \, dx[/tex]
The derivative is
y' = 5
Therefore
[tex]S = \int_{-1}^{3} \sqrt{1+25} \, dx =\sqrt{26}*(3-(-1))=20.396[/tex]
Note that
x = -1 +> y = 5(-1) - 4 = -9
x = 3 => y = 5(3) - 4 = 11
The distance between the points (-1, -9) and (3, 11) from the distance formula is
D = √[(3-(-1))² + (11-(-9))²] = √(16+400) = 20.396
This answer agrees with that obtained by integration.
Answer: 20.396
Obtained by integration and verified by the distance formula.
