Arc length of parametric curves

Three quick questions from earlier lessons.

In a tiny time $dt$ , a point moves $dx$ across and $dy$ up. How far does it travel?

Add up tiny hypotenuses.

Arc length of

x(t)

y(t)

for

a \le t \le b

L = \displaystyle\int_a^b \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2}\,dt

(if the curve is traced once).

$x = 3t$ , $y = 4t$ , $0 \le t \le 2$ .

x' = 3

y' = 4

Rates.

L = \displaystyle\int_0^2 5\,dt

√(9 + 16) = 5.

= 10

Check: from (0, 0) to (6, 8), distance 10.

Your turn: the circle $x = 2\cos t$ , $y = 2\sin t$ , $0 \le t \le 2\pi$ .

L = 4\pi

√(4 sin² + 4 cos²) = 2; ∫₀^{2π} 2 dt.

Find the length of the curve $x = 3\cos t$ , $y = 3\sin t$ , $0 \le t \le 2\pi$ .
Find the length of the curve $x = 5\cos t$ , $y = 5\sin t$ , $0 \le t \le 2\pi$ .

Find the length of the path $x = 3t$ , $y = 6t$ , $0 \le t \le 1$ . Give an exact value.
Find the length of the path $x = 4t$ , $y = 4t$ , $0 \le t \le 2$ . Give an exact value.

Three questions to check you have got it.

Find the length of the path $x = 3t$ , $y = 6t$ , $0 \le t \le 1$ . Give an exact value.
Find the length of the path $x = 4t$ , $y = 4t$ , $0 \le t \le 2$ . Give an exact value.
Find the length of the path $x = 2t$ , $y = 5t$ , $0 \le t \le 2$ . Give an exact value.

Arc length

$L = \displaystyle\int_a^b \sqrt{x'(t)^2 + y'(t)^2}\,dt$ .

Watch out: Pythagoras, not a sum.

End of the lesson