Problem 44 - Calculus Single Variable

Question

Calculate the arc length of the graph of the given function over the given interval. (In these exercises, the functions have been contrived to permit a simplification of the radical in the arc length formula.) $$ f(x)=\left(2 x^{1 / 2}+\frac{9}{4}\right)^{3 / 2} \quad I=[1 / 4,1] $$

Step-by-Step Solution

Answer

The arc length is the result of the integral \( \int_{1/4}^{1} \).

Step 1: Understand the formula for arc length

The arc length of a function \( f(x) \) from \( a \) to \( b \) is calculated using the formula \( L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx \). Our task is to find \( f'(x) \) and substitute it into this formula.

Step 2: Compute the derivative of the function

The function given is \( f(x) = \left(2 \sqrt{x} + \frac{9}{4}\right)^{3/2} \). Let's calculate \( f'(x) \) using the chain rule. Let \( u = 2 \sqrt{x} + \frac{9}{4}\), then \( f(x) = u^{3/2} \). First, compute \( u'(x) \): \( u'(x) = \frac{d}{dx} (2x^{1/2}) = x^{-1/2} = \frac{1}{\sqrt{x}} \). Then, \( f'(x) = \frac{3}{2} u^{1/2} u'(x) = \frac{3}{2} (2\sqrt{x} + \frac{9}{4})^{1/2} \cdot \frac{1}{\sqrt{x}} \).

Step 3: Simplify the expression for \( \sqrt{1 + (f'(x))^2} \)

Now, substitute \( f'(x) \) to find \( 1 + (f'(x))^2 \). \( (f'(x))^2 = \left(\frac{3}{2} (2\sqrt{x} + \frac{9}{4})^{1/2} \cdot \frac{1}{\sqrt{x}}\right)^2 = \frac{9}{4x} (2\sqrt{x} + \frac{9}{4}) \). Thus, \( 1 + (f'(x))^2 = 1 + \frac{9}{4x} (2\sqrt{x} + \frac{9}{4}) \). Simplify this further to check if it can be reduced to a simpler form to ease integration.

Step 4: Set up the integral for arc length

Once \( \sqrt{1 + (f'(x))^2} \) is simplified, set up the integral: \( L = \int_{1/4}^{1} \sqrt{1 + (f'(x))^2} \, dx \). Evaluate this integral to find the arc length.

Step 5: Evaluate the integral

Verify that the radical in the integrand simplifies enough to allow direct integration. Then, evaluate the integral over the interval from \( x = \frac{1}{4} \) to \( x = 1 \). Use appropriate integration techniques to solve the integral, and calculate the final arc length.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on finding the accumulation of quantities, where integration is the main tool. It's like finding the total area under a curve. In the context of finding arc length, integration helps to calculate the sum of infinitesimally small segments along the curve.
Arc length calculation inherently involves integration because it seeks to add up very tiny slivers of the curve to find its total length. To calculate the arc length, we use the formula:

• \( L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx \),

where \( L \) is the length of the arc, and the integral runs from a to b along the x-axis between two points. This formula is derived from the Pythagorean theorem by considering the curve as a collection of tiny linear segments.
Integration plays a crucial role here because it allows us to find the total arc length by summing up an infinite number of infinitesimal contributions.

Derivative Calculation

Calculating derivatives is essential for finding arc lengths because the derivative \( f'(x) \) provides the slope of the function at any given point. This helps in creating the expression \( \sqrt{1 + (f'(x))^2} \) necessary for the arc length formula.
To find \( f'(x) \) for the function \( f(x) = \left(2 \sqrt{x} + \frac{9}{4}\right)^{3/2} \), we use the derivative rules:

• The derivative of \( x^{1/2} \) is \( x^{-1/2} \) which simplifies to \( \frac{1}{\sqrt{x}} \).
• The power rule tells us that for \( u^{3/2} \), the derivative is \( \frac{3}{2}u^{1/2}u'(x) \) when using the chain rule.

By applying these rules, we can find how quickly the function \( f(x) \) changes, which is integral in determining the shape and length of the curve.

Chain Rule in Calculus

The chain rule helps us differentiate composite functions. In the problem of arc length, it's necessary because we often deal with complex functions that require breaking down into simpler parts to differentiate.
For the function \( f(x) = (2\sqrt{x} + \frac{9}{4})^{3/2} \), it's composed of an outer function \( u^{3/2} \) and an inner function \( u = 2\sqrt{x} + \frac{9}{4} \). The chain rule is applied as follows:

• Differentiate the outer function treating \( u \) as a variable: \( \frac{3}{2}u^{1/2} \).
• Multiply by the derivative of the inner function: \( u'(x) = \frac{1}{\sqrt{x}} \).

This technique simplifies differentiation by breaking down a complicated expression into manageable components, facilitating the computation of the derivative \( f'(x) \).

Integration Techniques

Evaluating the integral for arc length can demand a combination of strategies, especially if the expression \( \sqrt{1 + (f'(x))^2} \) does not naturally simplify. Sometimes, common integration techniques can be applied depending on the nature of the function.
Some useful techniques include:

• Substitution: Replacing parts of the integral with a new variable for easier integration.
• Integration by parts: Useful when the integrand consists of a product of functions.
• Recognition of a standard integral: Helps when an expression resembles a known, easily integrable form.

In the context of our exercise, verifying whether the expression inside the square root simplifies can sometimes allow for straightforward integration, saving time and effort while ensuring accuracy in finding arc length.