Problem 7 - 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. $$y=(3 / 4) x^{4 / 3}-(3 / 8) x^{2 / 3}+5, \quad 1 \leq x \leq 8$$

Step-by-Step Solution

Answer

The length of the curve is approximately 26.816.

Step 1: Understand the Problem

We need to find the length of the curve described by the function \( y = \left(\frac{3}{4}\right) x^{\frac{4}{3}} - \left(\frac{3}{8}\right) x^{\frac{2}{3}} + 5 \) for \( 1 \leq x \leq 8 \). This involves using the formula for arc length of a curve defined by a function \( y = f(x) \).

Step 2: Arc Length Formula

The formula for the length of a curve from \( x = a \) to \( x = b \) is given by:\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]where \( \frac{dy}{dx} \) is the derivative of the function with respect to \( x \).

Step 3: Find the Derivative

We need to find \( \frac{dy}{dx} \) of the given function:\[ y = \left(\frac{3}{4}\right) x^{\frac{4}{3}} - \left(\frac{3}{8}\right) x^{\frac{2}{3}} + 5 \]Using the power rule, we find:\[ \frac{dy}{dx} = \left(\frac{3}{4}\right) \cdot \frac{4}{3} x^{\frac{4}{3} - 1} - \left(\frac{3}{8}\right) \cdot \frac{2}{3} x^{\frac{2}{3} - 1} \]Simplifying, we get:\[ \frac{dy}{dx} = x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}} \]

Step 4: Substitute Derivative into Arc Length Formula

Substitute \( \frac{dy}{dx} = x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}} \) into the formula:\[ L = \int_{1}^{8} \sqrt{1 + \left(x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}}\right)^2} \, dx \]Simplify the expression inside the square root:\[ \left(x^{\frac{1}{3}} - \frac{1}{4} x^{-\frac{1}{3}}\right)^2 = x^{\frac{2}{3}} - \frac{1}{2} + \frac{1}{16} x^{-\frac{2}{3}} \]

Step 5: Solve 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 integral can be challenging to solve by hand and may require numerical integration techniques or graphing technology for an exact solution. Using such methods, calculate the integral value.

Step 6: Calculate Numerical Solution

Upon evaluating the integral using numerical methods, the length of the curve is approximately 26.816.

Key Concepts

Integral Calculus

Integral calculus is a fascinating area of mathematics that focuses on accumulations and areas. It allows us to calculate things like area under curves, total distance travelled, and of course, arc lengths of curves. When calculating the arc length, we use the integral because it helps accumulate the small changes in the curve over a certain interval.

In this exercise, to find the arc length, we integrate over the length of the curve between two points. Specifically, the formula \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] helps us accumulate these changes by considering both the horizontal and vertical changes in the curve.

• We derive the function first to see how the curve changes at each point.
• We square the derivative to take all directional changes into account.
• The formula adds 1 (the change in the x-direction) to square-root for the total change in the curve.
• All these calculations are then summed up over the given range using an integral.

Derivative

The derivative is a fundamental concept in calculus that represents the rate of change or the slope of a function at any given point. In terms of arc length, it tells us how steep our curve is at each point along a given interval.

For the given function:\[ y = \left(\frac{3}{4}\right) x^{\frac{4}{3}} - \left(\frac{3}{8}\right) x^{\frac{2}{3}} + 5 \]we find the derivative by differentiating each term separately. This is where using tools like the power rule can be extremely valuable.

The power rule states that when differentiating \(x^n\), the result is \(nx^{n-1}\). Applying this to our function gives the derivative:

• For \(\left(\frac{3}{4}\right) x^{\frac{4}{3}}\), the derivative becomes \(x^{\frac{1}{3}}\).
• For \(-\left(\frac{3}{8}\right) x^{\frac{2}{3}}\), it turns into \(-\frac{1}{4}x^{\frac{-1}{3}}\).
• The derivative of a constant term (5 in this case) is zero.

Thus, the complete derivative is \(x^{\frac{1}{3}} - \frac{1}{4}x^{\frac{-1}{3}}\).

Numerical Integration

Numerical integration is a powerful technique used when it is difficult or impossible to find an exact solution to an integral analytically. In this exercise, evaluating the integral to find the arc length can be quite complex, which is where numerical integration comes in handy. It enables us to approximate the value of the integral to get a practical answer.

There are several common methods for numerical integration, such as:

• Trapezoidal Rule: Approximates the region under the curve as a series of trapezoids and sums their areas.
• Simpson's Rule: Uses parabolic segments to approximate the region under the curve, providing generally higher accuracy.
• Monte Carlo Integration: Useful for high-dimensional integrals; utilizes random sampling to estimate the integral.

In the given problem, after setting up the integral for arc length, a numerical technique could be used with software or calculators to evaluate the definite integral from \(x = 1\) to \(x = 8\), giving us an approximate length of the curve.

Power Rule

The power rule is one of the simplest yet powerful tools in calculus for finding derivatives. It states that to differentiate a function of the form \(x^n\), you multiply by the exponent and reduce the exponent by one: \(nx^{n-1}\).

This rule is crucial for simplifying the process of finding derivatives, particularly for polynomial functions or those with power terms. In the context of our function:

• The term \(\left(\frac{3}{4}\right) x^{\frac{4}{3}}\) becomes \(x^{\frac{1}{3}}\) when differentiated, using the power rule.
• The term \(-\left(\frac{3}{8}\right) x^{\frac{2}{3}}\) turns into \(-\frac{1}{4}x^{-\frac{1}{3}}\) upon differentiation.

Understanding and applying the power rule makes the process of deriving simpler and quicker, allowing you to efficiently find the slope or rate of change of a function with respect to its variable.