8.2 Arc Length (Distance Traveled Along a Curve) in Parametric Form
If a curve $C$ is given by the parametric equations $x=f(t)$ and $y=g(t)$ such that $C$ does not intersect itself on the interval $a \leq t \leq b$ then the arc length $C$, or the distance traveled by a particle along the curve is given by
$L=\int_a^b \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^2} d t$
The displacement of a particle is the distance between its initial and final positions. The displacement of a particle between time $t=a$ and $t=b$ is given by
$\begin{aligned} \text { Dispalcement } & =\sqrt{\left[\int_a^b x^{\prime}(t) d t\right]^2+\left[\int_a^b y^{\prime}(t) d t\right]^2} \\ & =\sqrt{[x(b)-x(a)]^2+[y(b)-y(a)]^2} .\end{aligned}$
Example1
- A particle moves in the $x y$-plane so that its position at any time $t$, for $0 \leq t$, is given by $x(t)=e^t$ and $y(t)=2 \cos (t)$.
(a) Find the distance traveled by the particle from $t=0$ to $t=2$.
(b) Find the magnitude of the displacement of the particle between time $t=0$ and $t=2$.
▶️Answer/Explanation
Solution
(a) $\frac{d x}{d t}=e^t, \quad \frac{d y}{d t}=-2 \sin (t)$
$
\begin{aligned}
& \text { Distance traveled by the partic } \\
& =\int_0^2 \sqrt{\left.\left(e^t\right)^2+(-2 \sin t)\right)^2} d t \\
& \approx 7.035
\end{aligned}
$
Use a graphing calculator (in function mode) to find the value of the definite integral.
$
\begin{aligned}
(b) & \text { Displacement }=\sqrt{\left[\int_a^b x^{\prime}(t) d t\right]^2+\left[\int_a^b y^{\prime}(t) d t\right]^2} \\
& =\sqrt{\left[\int_0^2 e^t d t\right]^2+\left[\int_0^2(-2 \sin t) d t\right]^2} \\
& =\sqrt{(6.389)^2+(-2.832)^2} \quad \text { Use a graphing calculator. } \\
& \approx \sqrt{48.8395} \\
& \approx 6.988 \quad
\end{aligned}
$
Example 2
- The position of a particle at any time $t \geq 0$ is given by $x=t-t^2$ and $y=\frac{4}{3} t^{3 / 2}$. What is the total distance traveled by the particle from $t=1$ to $t=3$ ?
(A) 7.165
(B) 8.268
(C) 9.431
(D) 10.346
▶️Answer/Explanation
Ans:B
Example 3
- The position of particle at any time $t \geq 0$ is given by $x(t)=a(\cos t+t \sin t)$ and $y(t)=a(\sin t-t \cos t)$. What is the total distance traveled by the particle from $t=0$ to $t=\pi$ ?
(A) $\frac{1}{2} \pi a$
(B) $\pi a^2$
(C) $\frac{1}{2} \pi^2 a$
(D) $\frac{1}{2} \pi^2 a^2$
▶️Answer/Explanation
Ans:C
Example 4
- The length of the path described by the parametric equations $x=\sin t+\ln (\cos t)$ and $y=\cos t$, for $\frac{\pi}{6} \leq t \leq \frac{\pi}{3}$, is given by
(A) $\int_{\pi / 6}^{\pi / 3} \sqrt{\cos ^2 t+2 \sin t+2} d t$
(B) $\int_{\pi / 6}^{\pi / 3} \sqrt{\sin ^2 t+2 \cos t+2} d t$
(C) $\int_{\pi / 6}^{\pi / 3} \sqrt{\cot ^2 t+2 \cos t} d t$
(D) $\int_{\pi / 6}^{\pi / 3} \sqrt{\sec ^2 t-2 \sin t} d t$
▶️Answer/Explanation
Ans:D
Example 5
A particle moving in the $x y$ – plane has velocity vector given by $v(t)=\left\langle e^t-t, t \sin t\right\rangle$ for time $t \geq 0$. What is the magnitude of the displacement of the particle between time $t=0$ and $t=2$ ?
(A) 4.722
(B) 4.757
(C) 4.933
(D) 5.109
▶️Answer/Explanation
Ans:A