Problem 8 - 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. $$y=\left(x^{3} / 3\right)+x^{2}+x+1 /(4 x+4), \quad 0 \leq x \leq 2$$

Step-by-Step Solution

Answer

Use the arc length formula and numerical integration to compute the integral from 0 to 2.

Step 1: Understand the Arc Length Formula

The arc length for a curve defined by a function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by the formula: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]. Here, \( a = 0 \) and \( b = 2 \). We need to find the derivative \( \frac{dy}{dx} \) of the given function first.

Step 2: Differentiate the Function

Given the function \( y = \frac{x^3}{3} + x^2 + x + \frac{1}{4x+4} \), differentiate it with respect to \( x \). Applying the power rule to the polynomial terms, we get: \( \frac{dy}{dx} = x^2 + 2x + 1 \).For \( \frac{1}{4x+4} \), use the chain rule and quotient rule to get the derivative as\( -\frac{4}{(4x+4)^2} \).Combine these to get:\( \frac{dy}{dx} = x^2 + 2x + 1 - \frac{4}{(4x+4)^2} \).

Step 3: Simplify the Derivative

Calculate the expression \( \left(\frac{dy}{dx}\right)^2 \):\[ \left(\frac{dy}{dx}\right)^2 = \left(x^2 + 2x + 1 - \frac{4}{(4x+4)^2}\right)^2 \].Use algebraic expansion techniques and simplification tools to simplify the expression.

Step 4: Set Up the Integral for Arc Length

Insert the derivative squared into the arc length formula:\[ L = \int_{0}^{2} \sqrt{1 + \left(x^2 + 2x + 1 - \frac{4}{(4x+4)^2}\right)^2} \, dx \].

Step 5: Evaluate Arc Length Integral

This integral may involve complex manipulations or numerical integration techniques. A substitution method or numerical tools like graphing calculators might be needed to evaluate the integral accurately. After evaluation, this gives the length of the curve from \( x = 0 \) to \( x = 2 \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that helps us find the rate at which a function is changing at any given point. In essence, it's about finding the derivative, which tells us the slope or steepness of the curve at a specific point. For this exercise, we are differentiating the function

• \( y = \frac{x^3}{3} + x^2 + x + \frac{1}{4x+4} \),

with respect to \( x \). Each term is addressed:

• The first three terms are straightforward as they are polynomials, resulting in \( x^2 + 2x + 1 \).
• The last term, \( \frac{1}{4x+4} \), requires the chain rule and results in a slightly more complex calculation, \(-\frac{4}{(4x+4)^2}\).

When combined, we find the derivative to be \( x^2 + 2x + 1 - \frac{4}{(4x+4)^2} \). Differentiation, as shown, breaks down the function into simpler parts, making it easier to determine how each part contributes to changes in \( y \) as \( x \) varies. Understanding this change is key when dealing with curves in calculus.

Integral Calculus

Integral calculus helps us find quantities where the rate of change is known, such as finding the area under a curve or, in this case, the arc length of a curve. For arc length, the formula involves an integral with a function that incorporates the derivative:

• \[ L = \int_{0}^{2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]

Here, \( \left(\frac{dy}{dx}\right)^2 \) represents the square of the derivative, which captures how the function's slope contributes to the overall length of the curve from \( x = 0 \) to \( x = 2 \). The integral, therefore, accumulates these small segments of length over the interval \([0, 2]\), giving a comprehensive measurement of the curve's total length. Integral calculus thus allows us to extend the concept of summing up small changes, which is particularly useful in this problem where simple computations don't suffice and a continuous process like integration is needed.

Chain Rule

The chain rule is a powerful technique in differentiation used when dealing with composite functions. It's essential when our function of interest is formed by applying one function to another (e.g., nesting functions). In this exercise, it is particularly useful for differentiating the non-polynomial term,

• \( \frac{1}{4x+4} \).

The chain rule states that if you have a function \( g(x) = f(h(x)) \), then the derivative \( g'(x) \) is given by:

• \( g'(x) = f'(h(x)) \cdot h'(x) \).

For \( \frac{1}{4x+4} \), you set \( h(x) = 4x+4 \), making \( f(u) = \frac{1}{u} \) where \( u = h(x) \). By deriving each part, you apply the chain rule to find \(-\frac{4}{(4x+4)^2}\), capturing the curve's intricacy accurately. This demonstrates the chain rule's utility when faced with derivatives of more complex, layered expressions.

Numerical Integration

Numerical integration is a method used to approximate the value of integrals that are difficult or impossible to solve analytically. Especially for more complex integrals, such as the one we encounter when finding the arc length of a curve described by

• \( y = \frac{x^3}{3} + x^2 + x + \frac{1}{4x+4} \),

innovation through numerical tools becomes invaluable. These tools often include techniques like Simpson's Rule or the Trapezoidal Rule, which approximate the area under the curve using simpler geometric shapes or polynomial approximations.
For students without access to analytical solutions, graphing calculators or computer software can perform these numerical methods to find the integral's value. By dividing the interval into smaller segments, these methods enhance precision, enabling learners to get a more accurate measure of the curve's length without delving deeply into complex mathematics. They thus bridge the gap between theory and practical computation efficiently.