Finding Arc Lengths of Curves Given by Parametric Equations
Introduced| English |
|---|
| arc length/ɑːk leŋθ/ |
| speed/spiːd/ |
How far a parametric point travels
- A parametric point $\big(x(t),y(t)\big)$ moves through the plane. How long is the path it traces?
- That's the arc length 弧长 of the parametric curve, from $t=a$ to $t=b$.
- Each tiny time step moves the point by $dx=\tfrac{dx}{dt}\,dt$ across and $dy=\tfrac{dy}{dt}\,dt$ up.
- Add up those tiny hypotenuses to get the total length.
The formula
- The little step length is $\sqrt{dx^2+dy^2}=\sqrt{\big(\tfrac{dx}{dt}\big)^2+\big(\tfrac{dy}{dt}\big)^2}\,dt$.
- Integrating over the time interval:
-
$$L=\int_a^b \sqrt{\Big(\tfrac{dx}{dt}\Big)^2+\Big(\tfrac{dy}{dt}\Big)^2}\,dt$$
- Both component derivatives are squared, added, rooted, and integrated in $t$.
The arc length of a parametric curve is $\int_a^b(\ ?\ )\,dt$ where the integrand is...
Both derivatives squared, added, under a root.
For $x=t^2$, $y=t^3$: $(dx/dt)^2+(dy/dt)^2=$
$(2t)^2+(3t^2)^2=4t^2+9t^4$.
The integrand is the speed
- $\sqrt{\big(\tfrac{dx}{dt}\big)^2+\big(\tfrac{dy}{dt}\big)^2}$ is exactly the speed 速率 of the moving point.
- Speed is distance per unit time; integrating speed over time gives total distance.
- So arc length = the integral of speed — the parametric version of "distance = ∫ speed dt."
- This connects directly to motion problems (lesson 9.6).
Length is the integral of speed
y = ax^{1.5}
The parametric arc-length integrand is the point's speed — integrating speed over time gives the distance traveled.
The parametric arc-length integrand equals the point's ____.
Integral of speed = distance.
The parametric arc length is the same integral used for a particle's...
Integral of speed = distance traveled.
Setting it up in practice
- Compute $\tfrac{dx}{dt}$ and $\tfrac{dy}{dt}$, square and add, take the root.
- Integrate from the starting $t$ to the ending $t$ (the $t$-interval, not $x$ or $y$).
- Many of these integrals need a calculator; the setup is the graded skill.
- Keep both derivatives — dropping one loses the diagonal motion.
The limits of integration for parametric arc length are $t$-values.
Integrate over the time interval.
You can drop $\tfrac{dy}{dt}$ if $\tfrac{dx}{dt}$ is larger.
Both derivatives are needed for the diagonal motion.
The limits are $t$-values (the time interval), and both $\tfrac{dx}{dt}$ and $\tfrac{dy}{dt}$ appear, each squared under the root. Don't use $x$-limits, and don't forget one of the two derivatives. This is the same "speed integral" as distance traveled — not a signed quantity, so no absolute values needed (the root is already $\ge0$).
Find the length of $x=t^2,\ y=t^3$ for $0\le t\le 1$.
- $\tfrac{dx}{dt}=2t$, $\tfrac{dy}{dt}=3t^2$; sum of squares $=4t^2+9t^4=t^2(4+9t^2)$.
- $L=\displaystyle\int_0^1 \sqrt{t^2(4+9t^2)}\,dt=\int_0^1 t\sqrt{4+9t^2}\,dt$.
- $u$-sub $u=4+9t^2$: $=\tfrac{1}{27}\big(13^{3/2}-8\big)\approx1.44$.
The arc length of a parametric curve is $L=\int_a^b\sqrt{\big(\tfrac{dx}{dt}\big)^2+\big(\tfrac{dy}{dt}\big)^2}\,dt$ over the $t$-interval. The integrand is the point's speed, so this is the integral of speed = distance traveled. Square both component derivatives under one root.