Question

Use the Theorem of Pythagoras to find the length of the line segment
`x=t, y=5t(0\leq t\leq 1)`, and confirm that the value is consistent with the length computed using
`L=\int\limits^a_b \sqrt{ {((dx)/(dt) )^(2)+((dy)/(dt))^(2) }`

Answer 1

sqrt(26)

Explanation:

Length = sqrt[x² + y²]

t = 0,

(0, 0)

t = 1

(1 , 5)

Length = sqrt[1² + 5²]

= sqrt(1 + 25)

= sqrt(26)

Verification for consistency:

dx/dt = 1

dy/dt = 5

L = integral of [sqrt(1² + 5²)]

L = integral of sqrt(26)

L = t × sqrt(26)

Limits of t are 0 to 1

Upper limit - lower limit

sqrt(26) - 0

= sqrt(26)