Q. 3 - Calculus

Question

Mimic the argument in the reading for this section to argue that a reasonable definition for the arc length of a parametric curve $\left(x\right(t),y(t\left)\right)$ from $t=a$ to $t=b$ is

$\underset{n\to \mathrm{\infty }}{lim} \sum _{k=1}^n \sqrt{{\left(\mathrm{\Delta }{x}_k\right)}^2+{\left(\mathrm{\Delta }{y}_k\right)}^2},$

where $∆x_k$ and $∆y_k$ are defined as in the previous problem.

Step-by-Step Solution

Answer

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Step 1: Start from Riemann sum

The arc length is \( \lim_{n \to \infty} \sum \sqrt{(\Delta x_k)^2 + (\Delta y_k)^2} \).

Step 2: Convert to integral

Using \( \Delta x_k \approx x'(t_k)\Delta t \) and \( \Delta y_k \approx y'(t_k)\Delta t \):
Arc length = \( \int_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2}\,dt \).