Lengths of Curves (Rectification) Using Definite Integrals
Lengths of Curves (Rectification) Using Definite Integrals:

Objective:
 Calculate the length of a curve defined by a function $y=f(x)$over the interval $[a,b]$ using definite integrals.

Approach:
 The length of a curve over $[a,b]$ is approximated by dividing the interval into small segments and summing up their lengths. As the segments become infinitesimally small, the sum approaches the actual length.

Formula for Length of a Curve:
 The formula to find the length $L$ of a curve $y=f(x)$ over $[a,b]$ is given by: $L={\int}_{a}^{b}\sqrt{1+{\left(\frac{dy}{dx}\right)}^{2}}\text{\hspace{0.17em}}dx$
 Here, $\frac{dy}{dx}$ represents the derivative of $f(x)$ with respect to $x$.

Steps to Find Curve Length:

Step 1: Identify the Curve: Determine the curve represented by $y=f(x)$ and the interval $[a,b]$ for which the length is to be calculated.

Step 2: Derive the Differential Element: Calculate $\frac{dy}{dx}$ to find the differential element representing the length of a small segment of the curve.

Step 3: Setup Definite Integral: The length of the curve over $[a,b]$ is given by the definite integral formula: $L={\int}_{a}^{b}\sqrt{1+{\left(\frac{dy}{dx}\right)}^{2}}\text{\hspace{0.17em}}dx$

Step 4: Evaluate the Integral: Compute the definite integral to find the total length of the curve.


Example: Consider the curve $y=\sqrt{x}$ over the interval $[0,4]$. Let's find the length of this curve.

Steps to Find Curve Length:

Step 1: Identify the Curve and Interval: The curve $y=\sqrt{x}$ is defined over $[0,4]$.


Step 2: Derive the Differential Element: Calculate $\frac{dy}{dx}$ for $y=\sqrt{x}$: $\frac{dy}{dx}=\frac{1}{2\sqrt{x}}$

Step 3: Setup Definite Integral: Use the length formula: $L={\int}_{0}^{4}\sqrt{1+{\left(\frac{1}{2\sqrt{x}}\right)}^{2}}\text{\hspace{0.17em}}dx$Step 4: Evaluate the Integral: Compute the definite integral to find the curve length.

Result:
 After evaluating the integral, the length of the curve $y=\sqrt{x}$