Problem 16 - Calculus Early Transcendentals

Question

Find the arc length of the following curves on the given interval by integrating with respect to \(x\) $$y=\frac{2}{3} x^{3 / 2}-\frac{1}{2} x^{1 / 2} \text { on }[1,9]$$

Step-by-Step Solution

Answer

Question: Determine the arc length of the curve \(y = \frac{2}{3}x^{3/2} - \frac{1}{2}x^{1/2}\) on the interval \([1, 9]\). Answer: The arc length of the given curve on the interval \([1, 9]\) is equal to the integral: $$ L = \int_1^9 \sqrt{1 + \left(x^{1/2} - \frac{1}{4} x^{-1/2}\right)^2} dx $$

Step 1: Find the derivative of the function with respect to \(x\)#

Differentiating the given function \(y\) with respect to \(x\), we have: $$ y'(x) = \frac{d}{dx} \left(\frac{2}{3} x^{3/2} - \frac{1}{2} x^{1/2}\right) $$ Using the power rule, we get: $$ y'(x) = \frac{2}{3} \cdot \frac{3}{2}x^{1/2} - \frac{1}{2} \cdot \frac{1}{2} x^{-1/2} = x^{1/2} - \frac{1}{4} x^{-1/2} $$

Step 2: Square the derivative and add 1#

Now, we will square the derivative obtained in step 1 and add 1: $$ 1 + [y'(x)]^2 = 1 + \left(x^{1/2} - \frac{1}{4} x^{-1/2}\right)^2 $$

Step 3: Find the square root of the expression#

Find the square root of the expression obtained in step 2: $$ \sqrt{1 + [y'(x)]^2} = \sqrt{1 + \left(x^{1/2} - \frac{1}{4} x^{-1/2}\right)^2} $$

Step 4: Integrate the expression with respect to \(x\) on the interval \([1,9]\)#

We will now find the arc length \(L\) by integrating the expression obtained in step 3 with respect to \(x\) on the given interval \([1, 9]\): $$ L = \int_1^9 \sqrt{1 + \left(x^{1/2} - \frac{1}{4} x^{-1/2}\right)^2} dx $$ Unfortunately, this integral does not have a simple closed-form solution, so we would either have to approximate it using numerical integration methods or use a calculator or computer algebra system capable of calculating definite integrals.

Key Concepts

Derivative

In calculus, a derivative represents the rate at which a function changes with respect to an independent variable. Essentially, it gives us the slope of a function at any given point on its curve.
To find the derivative of a function, you apply a set of rules known as differentiation. These rules enable us to systematically find the rate of change of most functions.
In our exercise, the derivative of a given function \(y = \frac{2}{3} x^{3/2} - \frac{1}{2} x^{1/2}\) is calculated to understand how the curve of this function behaves on the interval from 1 to 9.

Power Rule

The power rule is a fundamental tool in calculus used for differentiating functions of the form \(x^n\). If you have a function \(y = x^n\), the derivative \(y'\) is \(nx^{n-1}\).
This rule simplifies the process of differentiation and is crucial when working with any polynomial functions.
For our function \(y = \frac{2}{3} x^{3/2} - \frac{1}{2} x^{1/2}\), applying the power rule allows us to find its derivative efficiently. Each term is treated separately:

• \(y' = \frac{2}{3} \cdot \frac{3}{2} x^{1/2}\) for \(x^{3/2}\)
• \(y' = -\frac{1}{2} \cdot \frac{1}{2} x^{-1/2}\) for \(x^{1/2}\)

This calculation results in the derivative \(y' = x^{1/2} - \frac{1}{4} x^{-1/2}\).

Integral

An integral is the reverse process of a derivative and is used to calculate areas under curves among other applications. Integration sums infinitely small data points, providing us with an aggregate value such as area, volume or in this case, arc length.
In our exercise, once we have formulated the expression for arc length, we switch our focus from differentiation to integration. The integral in the calculation of arc length is:
\(L = \int_1^9 \sqrt{1 + \left(x^{1/2} - \frac{1}{4} x^{-1/2}\right)^2} dx\).
This integral, which is within the interval [1,9], aims to calculate the arc length of the given function.

Numerical Integration

When an integral lacks a simple analytical form, like in our arc length problem, numerical integration provides a practical solution.
Numerical integration involves using algorithms to approximate the value of the integral. Common methods include the trapezoidal rule, Simpson's rule, or more sophisticated techniques implemented in software systems.
By breaking down the interval into smaller parts, numerical integration computes the sum that approximates the area under the curve. For our exercise, because the integral does not easily solve to a simple form, numerical methods become essential to approximate the arc length accurately.