Q. 3 - Calculus

Question

What is the definition of the arc length of a continuous function $f\left(x\right)$ on an interval $\left[a,b\right]$? Explain this definition in words in terms of a limit of sums, and explain the meaning of the notation used in the definition.

Step-by-Step Solution

Answer

Arc length = $\underset{n\to \infty }{\lim }\sum _{k=1}^n\sqrt{1+{\left(\frac{∆y_k}{∆x}\right)}^2}∆x$.

Step 1. Given Information.

A continuous function $f\left(x\right)$ on an interval $\left[a,b\right]$.

Step 2. Definition of arc length by limit of sum.

If the function is a differentiable and continuous derivative. Then the arc length of $f\left(x\right)$ is given by

$\underset{n\to \infty }{\lim }\sum _{k=1}^n\sqrt{1+{\left(\frac{∆y_k}{∆x}\right)}^2}∆x$

Here, $∆x=\frac{b-a}{n}$, $x_k=a+k∆x$, $∆y_k=f\left(x_k\right)-f(x_k-1)$ for all $k=0,1,2...., n$.