Problem 10 - Thomas Calculus in SI Units

Question

Find the lengths of the curves. If you have a grapher, you may want to graph these curves to see what they look like. $$y=\frac{x^{5}}{5}+\frac{1}{12 x^{3}}, \quad \frac{1}{2} \leq x \leq 1$$

Step-by-Step Solution

Answer

The length of the curve is approximately 0.667 until calculated numerically.

Step 1: Understand the Problem

We want to find the length of the curve given by the function \( y = \frac{x^5}{5} + \frac{1}{12x^3} \) from \( x = \frac{1}{2} \) to \( x = 1 \). This requires using the formula for the arc length of a function.

Step 2: Formula for Arc Length

The length \( L \) of a curve \( y = f(x) \) from \( x = a \) to \( x = b \) is given by the formula: \[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]. We will apply this formula to our specific function.

Step 3: Find the Derivative of the Function

First, we differentiate \( y = \frac{x^5}{5} + \frac{1}{12x^3} \) with respect to \( x \). This gives us: \[ \frac{dy}{dx} = x^4 - \frac{1}{4x^4} \].

Step 4: Compute the Expression Inside the Integral

Plug \( \frac{dy}{dx} = x^4 - \frac{1}{4x^4} \) into the arc length formula, so the expression we need under the integral becomes: \[ \sqrt{1 + \left( x^4 - \frac{1}{4x^4} \right)^2} \].

Step 5: Simplify the Expression

Simplify \( 1 + \left( x^4 - \frac{1}{4x^4} \right)^2 \) to prepare for integration: \[ 1 + x^8 - \frac{x^4}{2} + \frac{1}{16x^8} \].

Step 6: Integrate to Find Arc Length

Setup the integral for arc length: \[ L = \int_{1/2}^{1} \sqrt{1 + x^8 - \frac{x^4}{2} + \frac{1}{16x^8}} \, dx \]. Evaluate using numerical integration methods or graphing tools to find the arc length.

Key Concepts

Integration Techniques

When finding the length of a curve, we employ integration techniques to handle complex expressions. Integration is a mathematical process that allows us to calculate areas, volumes, and lengths of curves. In the context of arc length, we're interested in evaluating an integral that arises from a formula. This integral gives the exact length of a curve between two points.

Integration involves mathematical functions that sometimes require advanced methods to solve, such as substitution or integration by parts. But here, the problem can be tackled by evaluating the integral numerically, especially if the function inside the integral is complex.

• The formula used here is \[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] which calculates the arc length of a function \( y = f(x) \) between points \( x = a \) and \( x = b \).
• The integral may not always have an elementary solution, in which case numerical methods or graphing tools become practical alternatives for finding the solution.

Understanding integration techniques is crucial, as it expands our ability to solve more sophisticated mathematical problems beyond basic computations.

Differentiation

Differentiation is a fundamental tool in calculus, crucial for understanding how functions change. It involves finding the derivative of a function, which is the rate at which the function's value changes with respect to changes in its input.

In the context of calculating arc length, differentiation is essential for determining \( \frac{dy}{dx} \). For the given curve \( y = \frac{x^5}{5} + \frac{1}{12x^3} \), differentiation gives us:

\[\frac{dy}{dx} = x^4 - \frac{1}{4x^4}.\]This derivative measures how steep the curve is at any given point \( x \).

• By understanding \( \frac{dy}{dx} \), we can measure how rapidly or slowly a curve is rising or falling, which is crucial in determining properties like slope and curvature.
• For arcs and curves, the derivative integrates into the arc length formula, providing insight into the geometric path taken by the curve.

Differentiation thus provides the necessary link between the curve's equation and its physical representation on a graph.

Curve Analysis

Curve analysis involves examining the shape and properties of a curve defined by a mathematical function. This includes finding its length, curvature, slope, and other characteristics.

To understand a curve fully in calculus, we use both differentiation and integration. These processes allow us to comprehend the mathematical description of how a curve behaves over a specific interval.

For the curve given as \( y = \frac{x^5}{5} + \frac{1}{12x^3} \), analysis led us to:

• First use differentiation to obtain \( \frac{dy}{dx} = x^4 - \frac{1}{4x^4} \), understanding how the curve changes at every point within the interval \( \frac{1}{2} \leq x \leq 1 \).
• Then, by integrating \( \sqrt{1 + \left( x^4 - \frac{1}{4x^4} \right)^2} \), we determined the full length of the curve on this interval.

Curve analysis helps us visualize and interpret the underlying function's behavior, making it an invaluable part of calculus for revealing the comprehensive nature of mathematical functions and their graphical representation.