Lengths of Curves (Rectification) Using Definite Integrals

Lengths of Curves (Rectification) Using Definite Integrals:

1. Objective:

• Calculate the length of a curve defined by a function $y=f\left(x\right)$ over the interval $\left[a,b\right]$ using definite integrals.
2. Approach:

• The length of a curve over $\left[a,b\right]$ 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.
3. Formula for Length of a Curve:

• The formula to find the length $L$ of a curve $y=f\left(x\right)$ over $\left[a,b\right]$ 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 $f\left(x\right)$ with respect to $x$.
4. Steps to Find Curve Length:

• Step 1: Identify the Curve: Determine the curve represented by $y=f\left(x\right)$ and the interval $\left[a,b\right]$ 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 $\left[a,b\right]$ 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.

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

6. Steps to Find Curve Length:

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

7. 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.

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• Result:

• After evaluating the integral, the length of the curve $y=\sqrt{x}$ over $\left[0,4\right]$ is approximately $\frac{16}{3}$ units.