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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 $t$.

Each small piece of a curve is the hypotenuse of a dx-by-dy triangle

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).

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.