Problem 8 - Calculus Single Variable

Question

Calculate the arc length \(L\) of the graph of the given function over the given interval. $$ f(x)=(5-2 x)^{3 / 2} \quad I=[1 / 2,2] $$

Step-by-Step Solution

Answer

The arc length is approximately calculated using numerical methods for non-elementary integrals.

Step 1: Understand the Formula for Arc Length

The formula to calculate the arc length \( L \) of a function \( f(x) \) over an interval \( [a, b] \) is given by: \[ L = \int_a^b \sqrt{1 + \left( f'(x) \right)^2} \, dx \] Here, \( f'(x) \) is the derivative of \( f(x) \). We'll need this derivative for our calculations.

Step 2: Differentiate the Function

Given the function \( f(x) = (5 - 2x)^{3/2} \), we first need to find its derivative \( f'(x) \). Using the chain rule, let \( u = 5 - 2x \), then \( f(x) = u^{3/2} \) and \( f'(x) = \frac{d}{dx} u^{3/2} = \frac{3}{2}u^{1/2} \cdot (-2) \).Therefore, \[ f'(x) = \frac{3}{2} (5 - 2x)^{1/2} \cdot (-2) = -3(5 - 2x)^{1/2} \]

Step 3: Setup the Arc Length Integral

Substitute \( f'(x) \) into the arc length formula, giving: \[ L = \int_{1/2}^{2} \sqrt{1 + [-3(5 - 2x)^{1/2}]^2} \, dx \]Simplify the expression inside the square root: \[ \sqrt{1 + 9(5 - 2x)} \] Thus the integral becomes: \[ L = \int_{1/2}^{2} \sqrt{1 + 9(5 - 2x)} \, dx \]

Step 4: Simplify and Evaluate the Integral

Simplify the integrand: \[ \sqrt{1 + 9 \cdot (5 - 2x)} = \sqrt{1 + 45 - 18x} = \sqrt{46 - 18x} \]Now, evaluate the integral: \[ L = \int_{1/2}^{2} \sqrt{46 - 18x} \, dx \]This integral can be evaluated using substitution or numerical methods as it doesn't have an elementary antiderivative. Using a basic integration technique or software/numerical approximation, we find the arc length approximately.

Key Concepts

Derivative Calculation

The process of finding a derivative is crucial in calculus and especially important for problems involving rates of change or slope calculations. When we have a function, like the one provided in the exercise, \[ f(x) = (5 - 2x)^{3/2} \],we aim to find its derivative, which tells us how the function's value changes with a slight change in \( x \). This step is foundational in determining the arc length of the given function.

• The derivative of a power function is found using the power rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \).
• The chain rule, which we'll discuss separately, often plays a major role in complex derivatives.

Applying these rules, the derivative of \( f(x) \) becomes essential for the arc length formula. The derivative gives us \( f'(x) \), allowing the next steps in arc length calculation.

Definite Integral

In calculus, the definite integral is used to calculate the area under a curve between two limits, \( a \) and \( b \). For arc length calculation, it helps to sum infinitely small line segments to form the arc between \( a \) and \( b \). This is represented by the integral symbol \( \int \), and it represents a continuous sum. For our problem, the arc length \( L \) is computed using the formula: \[ L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx \] The steps generally include:

• Substituting the derivative \( f'(x) \) into the formula.
• Determining the limits of integration, which in this case are \( 1/2 \) to \( 2 \).

The solution involves simplifying the resulting integrand and evaluating the integral either analytically or using numerical methods if direct integration isn't feasible. This calculation provides the total length of the curve over the specified interval.

Chain Rule in Differentiation

The chain rule is a fundamental principle in calculus used to differentiate compositions of functions. When a function is composed of multiple layers, the chain rule allows us to "peel back" each layer to find the derivative. This is especially important for derivations like the one in our exercise. For the function \( f(x) = (5 - 2x)^{3/2} \), the inner function \( u(x) = 5 - 2x \) is encapsulated within an outer function, \( u^{3/2} \). Consequently, the chain rule applies as follows:

• Take the derivative of the outer function with respect to the inner function, yielding the term \( \frac{3}{2} u^{1/2} \).
• Then, multiply by the derivative of the inner function, \( -(2) \).

The result is \( f'(x) = -3(5 - 2x)^{1/2} \). Using the chain rule simplifies complex functions into manageable calculations, ensuring we can effectively apply further methods like the arc length formula.