Calculating the Arc Length of a Curve: Y ( frac{x^3}{6} - frac{1}{2x} ) from x 1 to x 2

Introduction

In this article, we will explore how to calculate the arc length of a curve represented by the function y ( frac{x^3}{6} - frac{1}{2x} ) from x 1 to x 2. This process involves using integral calculus and will help us understand the geometric properties of the curve.

Step-by-Step Calculation

To begin, we need to find the derivative of the given function with respect to x to determine the slope of the tangent at any point on the curve. The function is given as:

y ( frac{x^3}{6} - frac{1}{2x} )

First, let's calculate the derivative y':

1. ( y' frac{d}{dx} ( frac{x^3}{6} - frac{1}{2x} ) )

2. ( y' frac{x^2}{2} - frac{-1}{2x^2} )

3. Simplifying, we get:

y' ( frac{x^2}{2} frac{1}{2x^2} ) )

Next, using the arc length formula, we need to integrate the square root of ( 1 (y')^2 ) from x 1 to x 2.

Integrating the Arc Length

1. L int_{1}^{2} sqrt{1 (y')^2} dx )

2. Substitute y' into the formula:

L int_{1}^{2} sqrt{1 ( frac{x^2}{2} frac{1}{2x^2} )^2} dx )

3. Simplify the expression inside the square root:

L int_{1}^{2} sqrt{1 frac{x^4 1}{2x^2} frac{1}{4x^4}} dx )

4. Combine the terms under the square root:

L int_{1}^{2} sqrt{frac{x^8 2x^4 1 2x^4 - 2x^4 1}{4x^4}} dx )

5. Further simplify:

L int_{1}^{2} sqrt{frac{x^8 2x^4 1}{4x^4}} dx )

6. Simplify the square root:

L int_{1}^{2} frac{x^4 1}{2x^2} dx )

7. Split the integral into two parts:

L int_{1}^{2} frac{x^2}{2} dx - int_{1}^{2} frac{1}{2x^2} dx )

8. Evaluate each part separately:

L left[ frac{x^3}{6} right]_{1}^{2} - left[ frac{-1}{2x} right]_{1}^{2} )

9. Substitute the limits of integration:

L left( frac{2^3}{6} - frac{-1}{2 cdot 2} right) - left( frac{1^3}{6} - frac{-1}{2 cdot 1} right) )

10. Simplify the expression:

L left( frac{8}{6} frac{1}{4} right) - left( frac{1}{6} frac{1}{2} right) )

11. Combine the terms:

L left( frac{4}{3} frac{1}{4} right) - left( frac{1}{6} frac{1}{2} right) )

12. Find a common denominator and simplify:

L left( frac{16}{12} frac{3}{12} right) - left( frac{2}{12} frac{6}{12} right) )

13. Combine the fractions:

L frac{19}{12} - frac{8}{12} frac{17}{12} )

Conclusion

Thus, the arc length L of the curve given by y ( frac{x^3}{6} - frac{1}{2x} ) from x 1 to x 2 is frac{17}{12}.

Further Reading and Examples

For more examples and detailed explanations on how to calculate arc length and other calculus problems, please refer to the Calculus II: Arc Length tutorial.