Problem 14 - Calculus: Early Transcendentals

Question

Find the arc length of the function on the given interval. $$ f(x)=2 x^{3 / 2}-\frac{1}{6} \sqrt{x} \text { on }[0,9] $$

Step-by-Step Solution

Answer

The arc length is found by evaluating \(\int_0^9 \sqrt{9x + \frac{1}{2} + \frac{1}{144x}} \, dx\) numerically.

Step 1: Understand the formula for arc length

The arc length of a curve defined by a function \(y = f(x)\) over the interval \([a, b]\) is given by the integral formula: \[ L = \int_a^b \sqrt{1 + \left(f'(x)\right)^2} \, dx \]. To find the arc length of the given function, we first need to compute its derivative, \(f'(x)\).

Step 2: Find the derivative of the function

Given \(f(x) = 2x^{3/2} - \frac{1}{6}\sqrt{x}\), we find its derivative \(f'(x)\). Using power rules: \[ f'(x) = \frac{d}{dx}(2x^{3/2}) - \frac{d}{dx}\left(\frac{1}{6}x^{1/2}\right) \]\(f'(x) = 2 \times \frac{3}{2}x^{1/2} - \frac{1}{6} \times \frac{1}{2}x^{-1/2}\)Simplify to get:\[ f'(x) = 3x^{1/2} - \frac{1}{12}x^{-1/2} \].

Step 3: Set up the integral for arc length

Substitute \(f'(x)\) into the arc length formula:\[ L = \int_0^9 \sqrt{1 + \left(3x^{1/2} - \frac{1}{12}x^{-1/2}\right)^2} \, dx \].Simplify inside the square root to prepare for integration.

Step 4: Simplify the expression under the square root

Calculate the square \((3x^{1/2} - \frac{1}{12}x^{-1/2})^2\):\[(3x^{1/2})^2 - 2 \times 3x^{1/2} \times \frac{1}{12}x^{-1/2} + \left(\frac{1}{12}x^{-1/2}\right)^2 \]\[= 9x + \frac{1}{144x} - \frac{1}{2} \].Substitute into the integral to get:\[ L = \int_0^9 \sqrt{1 + 9x - \frac{1}{2} + \frac{1}{144x}} \, dx \]\[= \int_0^9 \sqrt{9x + \frac{1}{2} + \frac{1}{144x}} \, dx \].

Step 5: Evaluate the integral numerically or use a suitable method

The integral \(\int_0^9 \sqrt{9x + \frac{1}{2} + \frac{1}{144x}} \, dx\) may be complex for simple algebraic solutions, so it's best solved using numerical techniques or software solutions suited for definite integrals. Approximations through numerical methods give us the arc length.

Key Concepts

Derivative of Function

The derivative of a function is a key concept in calculus. It helps us understand how a function changes at any given point. For a function like \( f(x) = 2x^{3/2} - \frac{1}{6}\sqrt{x} \), computing its derivative means finding \( f'(x) \), which represents the rate of change of the function with respect to \( x \). To calculate this, we employ the power rule for derivatives. The power rule states that if you have a function of the form \( x^n \), its derivative is \( nx^{n-1} \).

Using this rule, for our function, \( f'(x) \) is calculated as follows:

• The derivative of \( 2x^{3/2} \) is \( 3x^{1/2} \) because you multiply the exponent by the coefficient and reduce the exponent by 1.
• The derivative of \( -\frac{1}{6}\sqrt{x} \) becomes \( -\frac{1}{12}x^{-1/2} \), applying the rule to \( x^{1/2} \).

Thus, the derivative of the function is \( f'(x) = 3x^{1/2} - \frac{1}{12}x^{-1/2} \). This derivative is crucial for calculating the arc length of the function.

Integral for Arc Length

Finding the arc length of a curve defined by \( y = f(x) \) over an interval is a fundamental application of calculus. The arc length \( L \) is determined using the integral formula: \[ L = \int_a^b \sqrt{1 + \left(f'(x)\right)^2} \, dx \].
This formula takes into account the rate of change of the function as represented by its derivative \( f'(x) \). By summing up these small lengths incrementally over the interval, we get the total arc length of the curve.

For the given function and the derivative, the arc length integral becomes: \[ L = \int_0^9 \sqrt{1 + \left(3x^{1/2} - \frac{1}{12}x^{-1/2}\right)^2} \, dx \].
Substituting the derivative into the arc length formula helps us proceed towards solving for the desired length.

Power Rule for Derivatives

The power rule is a fundamental tool in calculus that simplifies differentiation. It's specially used when dealing with functions involving polynomial expressions or powers. Given a function \( x^n \), its derivative can be easily found using the power rule: \( nx^{n-1} \).

This simplicity allows quick calculations and is especially useful in functions where multiple such terms are involved, as is the case with our given function. Each term of the function \( f(x) = 2x^{3/2} - \frac{1}{6}\sqrt{x} \) is differentiated separately:

• The term \( 2x^{3/2} \) becomes \( 3x^{1/2} \) after applying the power rule.
• The term \( -\frac{1}{6}x^{1/2} \) simplifies to \(-\frac{1}{12}x^{-1/2} \).

By applying the power rule, finding derivatives becomes straightforward, making further calculations, such as integral setups for arc length, more manageable.

Numerical Integration Techniques

Sometimes, even with the best algebraic techniques, evaluating an integral can be daunting. Numerical integration techniques offer a way to approximate these integrals, making them achievable when analytical solutions are complex. Popular methods include Simpson's Rule, Trapezoidal Rule, and using computational software.

In our exercise, the integral \( \int_0^9 \sqrt{9x + \frac{1}{2} + \frac{1}{144x}} \, dx \) proves challenging for symbolic integration. Here, we turn to numerical methods to approximate the arc length:

• **Simpson's Rule**: Approximates the integral by dividing the region into even intervals and fitting parabolas through the data points.
• **Trapezoidal Rule**: Breaks the area under the curve into trapezoids for approximation rather than rectangles or other shapes.
• **Computational Tools**: Use software or calculators for precision when manual methods are cumbersome.

These methods ensure that even complex integrals can be solved accurately within reasonable margins of error.