Q. 2 - Calculus

Question

To approximate the arc length of a parametric curve on

some interval from $t=a$ to $t=b$, we subdivide $[a,b]$

into n pieces to define $t_1=a,t_2,t_3,.....,t_n.$ and consider the line segments connecting $\left(x\right(t_k-1),y(t_k-1\left)\right)$ and $\left(x\right(t_k),y(t_k\left)\right)$ for each $k=1,2,.....,n.$ Define $∆x_k$

and $∆y_k$ for the kth line segment in terms of the coordinates shown in the following figure:

Step-by-Step Solution

Answer

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Step 1: Describe the approximation

Subdivide \( [a, b] \) into \( n \) pieces: \( t_1 = a, t_2, \ldots, t_n \). Connect consecutive points \( (x(t_k), y(t_k)) \) with line segments.

Step 2: Sum the distances

Arc length \( \approx \sum_{k=1}^{n-1} \sqrt{(x(t_{k+1})-x(t_k))^2 + (y(t_{k+1})-y(t_k))^2} \). As \( n \to \infty \), this approaches \( \int_a^b \sqrt{(x'(t))^2 + (y'(t))^2}\,dt \).