Q. 4 - Calculus

Question

Suppose $f\left(x\right)$ is a differentiable function with a continuous derivative. Describe the arc length of $f\left(x\right)$ on the interval $\left[a,b\right]$ in terms of a definite integral. Why is this definite integral equal to the definition of arc length for $f\left(x\right)$ on [a, b] ? 

Step-by-Step Solution

Answer

Arc length =${\int }_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}dx$.

Step 1. Given Information.

The function $f\left(x\right)$ is a differentiable and continuous derivative on $\left[a,b\right]$.

Step 2. Explanation.

The arc length of $f\left(x\right)$ from x = a to x = b can be represented by the definite integral

${\int }_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}dx$

The length of the curve traced out by the graph of a function $f\left(x\right)$ on an interval $\left[a,b\right]$ is the same as the area under the graph of $\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}$ with interval $\left[a,b\right]$.