Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the length of the indicated curve. \(y=4 x^{3 / 2}\) between \(x=1 / 3\) and \(x=5\)

Step-by-Step Solution

Answer

The length of the curve is approximately 44.334 units.

Step 1: Understand the Formula for Arc Length

The formula for the length of a curve expressed as a function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]Identify that we need to find \( \frac{dy}{dx} \) for the function \( y = 4x^{3/2} \).

Step 2: Differentiate the Function

Find the derivative \( \frac{dy}{dx} \) of the function \( y = 4x^{3/2} \).Using the power rule on \( x^{3/2} \), we get \[\frac{dy}{dx} = 4 \cdot \frac{3}{2}x^{1/2} = 6x^{1/2}\]

Step 3: Set Up the Integrand

Substitute \( \frac{dy}{dx} = 6x^{1/2} \) into the arc length formula:\[L = \int_{1/3}^{5} \sqrt{1 + (6x^{1/2})^2} \, dx\]This simplifies to:\[L = \int_{1/3}^{5} \sqrt{1 + 36x} \, dx\]

Step 4: Simplify and Compute the Integral

We now compute the integral\[L = \int_{1/3}^{5} \sqrt{1 + 36x} \, dx\]To solve this, use the substitution \( u = 1 + 36x \), thus \( du = 36 \, dx \), or \( dx = \frac{du}{36} \). Convert the limits accordingly:- When \( x = 1/3 \), \( u = 13 \).- When \( x = 5 \), \( u = 181 \).The integral becomes:\[L = \int_{13}^{181} \frac{1}{36} \sqrt{u} \, du\]Further simplifying results in:\[L = \frac{1}{36} \int_{13}^{181} u^{1/2} \, du\]

Step 5: Integrate

Solve the integral:\[L = \frac{1}{36} \int u^{1/2} \, du = \frac{1}{36} \left[ \frac{2}{3} u^{3/2} \right]_{13}^{181}\]Evaluating this gives:\[L = \frac{1}{36} \cdot \frac{2}{3} \left[u^{3/2} \big|_{13}^{181}\right]\]

Step 6: Calculate the Definite Integral Limits

Evaluate \( u^{3/2} \) at the specified limits:\[L = \frac{1}{54} \left((181)^{3/2} - (13)^{3/2}\right)\]Calculate:\[(181)^{3/2} = 181 \times \sqrt{181},\quad (13)^{3/2} = 13 \times \sqrt{13}\]Find these values using a calculator or numerical estimation.

Step 7: Final Calculation

Calculate the values:\[181^{3/2} \approx 2440.84,\quad 13^{3/2} \approx 46.77\]Substitute back into the equation:\[L = \frac{1}{54} \left( 2440.84 - 46.77 \right)\]Finally:\[L \approx \frac{1}{54} \times 2394.07 \approx 44.334\]Thus, the length of the curve is approximately 44.334.

Key Concepts

Differentiation

Differentiation helps us find the rate at which a function changes at any given point, which is crucial for understanding how curves behave. In this exercise, we need to differentiate the function given by \( y = 4x^{3/2} \). This derivative, \( \frac{dy}{dx} \), tells us how the slope of the tangent to the curve changes as \( x \) changes.

One of the most powerful tools in differentiation is the Power Rule, which makes it straightforward to differentiate terms like \( x^{n} \). According to the Power Rule, the derivative of \( x^{n} \) is \( nx^{n-1} \). Using this rule, we quickly find that \( \frac{dy}{dx} = 6x^{1/2} \), showing how the slope changes as x increases.

Integration

Integration is the process of finding the accumulation of quantities, which in this context, helps us calculate the total arc length of a curve. Unlike differentiation which gives us a rate of change, integration allows us to sum these instantaneous changes over a specified interval, thus giving us the arc length of the function between two points.

The integral we're dealing with in this exercise is designed to find the arc length of a specific function from \( x = 1/3 \) to \( x = 5 \). We set up the integral of a function that includes \( \sqrt{1 + (6x^{1/2})^2} \). This expression arises from combining the slope \( \frac{dy}{dx} \) with a constant '1' to account for the straight line portion of each infinitesimal section of the curve.

Power Rule

The Power Rule is key in both differentiation and integration. While it simplifies the differentiation process, it is equally important in integration where it helps in reversing the differentiation process. This rule states that the integral of \( x^{n} \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

In our arc length calculation, we used the Power Rule to simplify our integrand \( u^{1/2} \), applying it as \( \frac{u^{3/2}}{3/2} \). Notice that in definite integrals, you do not add the constant of integration \( C \), as the limits of integration will account for any constant.

Definite Integral

Definite integrals give us the total value accumulated by a function over a certain interval, from point \( a \) to \( b \). They are represented as \( \int_{a}^{b} f(x) \, dx \). For arc length, this translates to finding \( \int_{1/3}^{5} \sqrt{1 + (6x^{1/2})^2} \, dx \), which calculates the total length of the curve in that interval.

In our solution, we perform a substitution to make the integral more manageable. We set \( u = 1 + 36x \), transforming the integral into \( \int_{13}^{181} \frac{1}{36} u^{1/2} \, du \). By simplifying and evaluating this expression, we find the curve's length. Definite integrals provide a method to arrive at a numeric answer, specifically that the length of our arc is approximately 44.334.