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(x) over the interval [a,b] using definite integrals.
  2. 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.
  3. 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=ab1+(dydx)2dx
    • Here, dydx represents the derivative of f(x) with respect to x.
  4. 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 dydx 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=ab1+(dydx)2dx

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

  5. Example: Consider the curve y=x over the interval [0,4]. Let's find the length of this curve.

  6. Steps to Find Curve Length:

    • Step 1: Identify the Curve and Interval: The curve y=x is defined over [0,4].

  7. Step 2: Derive the Differential Element: Calculate dydx for y=x: dydx=12x

  • Step 3: Setup Definite Integral: Use the length formula: L=041+(12x)2dxStep 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=x over [0,4] is approximately 163 units.