Calculating the Arc Length of a Curve Using Integral Calculus

Understanding how to calculate the arc length of a curve is a fundamental concept in integral calculus. In this article, we will provide a detailed explanation on how to find the exact value of the arc length of the curve defined by the equation y x^3 / 6 - 1/2x from x 1 to x 2. The process involves the application of the arc length formula and step-by-step integration.

1. The Equation and Its Derivative

The given curve is represented by the equation:

y x^3 / 6 - 1/2x

First, we need to find the derivative of the function y with respect to x. Let's start by rewriting the function in a more manageable form:

y x^3 / 6 - x^{-1}

The derivative y' is given by:

y' x^2 / 2 - (-1)(x^{-2}) x^2 / 2 1 / 2x^2

Combining the terms, we get:

y' (x^4 - 1) / 2x^2

2. Arc Length Formula

The formula for the arc length L of a curve from x a to x b is expressed as:

L ∫ab√(1 (y')2) dx

Substituting our derivative y' into the formula, we have:

L ∫12√(1 ((x^4 - 1) / 2x^2)2) dx

3. Simplifying the Integral

Let's simplify the expression under the square root:

L ∫12√(1 (x^4 - 1) / 4x^4) dx

We can bring the terms over a common denominator:

L ∫12√((4x^4 x^8 - 2x^4 - 1) / 4x^4) dx

This simplifies to:

L ∫12√((x^8 2x^4 1) / 4x^4) dx

Recognizing that x^8 2x^4 1 is a perfect square, we can rewrite it:

L ∫12√((x^4 1) / 4x^2) dx

This further simplifies to:

L ∫12(x^4 / (4x^2) 1 / (4x^2)) / 2 dx

Which simplifies to:

L ∫12(x^2 / 2 1 / 2x^2) dx

4. Integration to Find the Arc Length

Now, we can integrate the simplified expression:

L [x^3 / 6 - 1 / 2x] 12

Evaluating this at the limits of integration:

L (2^3 / 6 - 1 / (2*2)) - (1^3 / 6 - 1 / (2*1))

This gives us:

L (8 / 6 - 1 / 4) - (1 / 6 - 1 / 2)

Simplifying further:

L (16 / 12 - 3 / 12) - (2 / 12 - 6 / 12)

Which results in:

L 13 / 12 - (-4 / 12) 17 / 12

Conclusion

Through the application of the arc length formula and integration, we have determined that the arc length of the curve y x^3 / 6 - 1 / 2x from x 1 to x 2 is equal to 17 / 12. This example serves as a practical demonstration of how integral calculus can be used to solve real-world mathematical problems.

Resources

For more detailed examples and further explanation on calculating arc length in integral calculus, you can explore the following resource:

Calculus II - Arc Length