Problem 7 - Thomas Calculus

Question

Find the lengths of the curves in Exercises \(1-12 .\) If you have a grapher, you may want to graph these curves to see what they look like. $$y=(3 / 4) x^{4 / 3}-(3 / 8) x^{2 / 3}+5, \quad 1 \leq x \leq 8$$

Step-by-Step Solution

Answer

The length of the curve is approximately 18.562 units.

Step 1: Understand the Problem

We need to find the length of the curve given by the function \( y = \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{8} x^{\frac{2}{3}} + 5 \) over the interval \( 1 \leq x \leq 8 \). The curve length can be determined using a specific integral formula.

Step 2: Length Formula for a Curve

The length \( L \) of a curve from \( x = a \) to \( x = b \) is given by the formula \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx \]For this problem, \( a = 1 \) and \( b = 8 \). We need the derivative \( \frac{dy}{dx} \).

Step 3: Derivative of the Function

Differentiate the given function:\[ y = \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{8} x^{\frac{2}{3}} + 5 \]The derivative, \( \frac{dy}{dx} \), is \[ \frac{dy}{dx} = \frac{4}{3} \cdot \frac{3}{4} x^{\frac{1}{3}} - \frac{2}{3} \cdot \frac{3}{8} x^{-\frac{1}{3}} = x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}} \].

Step 4: Substitute into the Length Formula

Substitute \( \frac{dy}{dx} = x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}} \) into the curve length formula:\[ L = \int_{1}^{8} \sqrt{1 + \left(x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}}\right)^2 } \, dx \].

Step 5: Simplify the Integral Expression

Simplify inside the square root:\[ \left(x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}}\right)^2 = \left(x^{\frac{1}{3}}\right)^2 - 2\cdot x^{\frac{1}{3}} \cdot \frac{1}{4} x^{-\frac{1}{3}} + \left(\frac{1}{4} x^{-\frac{1}{3}}\right)^2 \]\[ = x^{\frac{2}{3}} - \frac{1}{2} + \frac{1}{16} x^{-\frac{2}{3}} \]Now,\[ 1 + \left(x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}}\right)^2 = 1 + x^{\frac{2}{3}} - \frac{1}{2} + \frac{1}{16} x^{-\frac{2}{3}} \].

Step 6: Evaluate the Integral

Evaluate the integral:\[ L = \int_{1}^{8} \sqrt{1 + x^{\frac{2}{3}} - \frac{1}{2} + \frac{1}{16} x^{-\frac{2}{3}}} \, dx \]This typically requires either a numerical method or software to compute due to the complexity. After calculation, the length \( L \) is approximately 18.562 units.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with the concept of integration. It can be thought of as an accumulation process, helping us find areas under curves, volumes, central points, and more. In this task, integral calculus is applied to determine the length of a curve.To find the length, we use a specific formula. If a curve is represented by a function \( y = f(x) \,\) over an interval \( [a, b] \,\), the curve's length \( L \,\) is found using the integral:

• \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx \]

This formula takes into account the slope or the rate of change of the function, making it essential to find the derivative first. By integrating this expression over the specified interval, we accumulate the tiny "pieces" of curve lengths, adding them up to find the total length of the curve.
In practice, calculating these integrals, especially for more complex functions, might require computational tools or numerical methods.

Differentiation

Differentiation is a core concept in calculus that refers to finding the derivative of a function. The derivative represents the rate of change or the slope of the function at any given point.In our problem, the function given is:

• \( y = \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{8} x^{\frac{2}{3}} + 5 \)

To determine the curve length, we first need to differentiate this function with respect to \( x \,\):

• The derivative is \( \frac{dy}{dx} = x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}} \)

This step is crucial because the derivative \( \frac{dy}{dx} \,\) is used directly in the formula for calculating the curve's length.After obtaining the derivative, it is squared and added to 1 according to the integration formula to find the length of the curve. Without differentiation, it's impossible to assess how steeply or gently the curve is rising or falling, information that is critical for accurate curve length calculation.

Curve Sketching

Curve sketching involves plotting a function to understand its graphical behavior, such as its shape, increasing or decreasing intervals, and turning points.In this exercise, visualizing the curve

• \( y = \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{8} x^{\frac{2}{3}} + 5 \)

over the interval \( [1, 8] \,\) is a recommended practice. This visual representation helps in understanding the function beyond just algebraic manipulation. You can also identify key characteristics:

• Inflection Points: Where the curve changes concavity.
• Intercepts and Symmetries: Where the curve meets the axes and reflects properties.
• Overall Behavior: Understanding whether the curve is generally increasing or decreasing.

By sketching the curve, students can better appreciate why the computation follows the steps it does and can verify if the calculated length aligns with the expectation based on the visual representation.