Problem 2 - Thomas Calculus in SI Units

Question

Find the lengths of the curves. If you have a grapher, you may want to graph these curves to see what they look like. $$y=x^{3 / 2} \quad \text { from } \quad x=0 \text { to } x=4$$

Step-by-Step Solution

Answer

The length of the curve is approximately \( \frac{8}{27} (10\sqrt{10} - 1)\).

Step 1: Understand the Problem

We need to find the length of the curve given by the function \(y = x^{3/2}\) over the interval from \(x = 0\) to \(x = 4\). This is a calculus problem that requires the use of the arc length formula.

Step 2: Recall the Arc Length Formula

The length of a curve \( y=f(x) \) from \( x=a \) to \( x=b \) is given by the formula: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]. We will apply this formula to find the length of the curve.

Step 3: Compute the Derivative \( \frac{dy}{dx} \)

Find the derivative of \( y = x^{3/2} \). Using the power rule, \( \frac{dy}{dx} = \frac{3}{2}x^{1/2} \).

Step 4: Set Up the Arc Length Integral

Substitute \( \frac{dy}{dx} = \frac{3}{2}x^{1/2} \) into the arc length formula. The integral becomes: \[ L = \int_{0}^{4} \sqrt{1 + \left( \frac{3}{2}x^{1/2} \right)^2} \, dx \].

Step 5: Simplify the Integrand

Inside the square root, we simplify: \( \left( \frac{3}{2}x^{1/2} \right)^2 = \frac{9}{4}x \). Thus, the integrand becomes \( \sqrt{1 + \frac{9}{4}x} \).

Step 6: Calculate the Integral

The integral \( \int_{0}^{4} \sqrt{1 + \frac{9}{4}x} \, dx \) can be calculated using substitution or other integration techniques. Let \( u = 1 + \frac{9}{4}x \), then \( du = \frac{9}{4} \, dx \). Solving gives: \[ L = \frac{4}{9} \left[ \frac{2}{3}\left(1 + \frac{9}{4}x\right)^{3/2} \right]_{0}^{4} \].

Step 7: Evaluate the Definite Integral

Plug in the limits 0 and 4: \( \frac{4}{9} \left[ \frac{2}{3}(10^{3/2} - 1^{3/2}) \right] \). Compute \( 10^{3/2} = 10 \sqrt{10} \) and subtract 1. Simplify to find \(L = \frac{8}{27} (10\sqrt{10} - 1)\).

Step 8: Compute the Final Length

Evaluate \(L\) with the calculated values to get the numerical length of the curve.

Key Concepts

Arc Length Formula

The arc length formula is a crucial concept in calculus for determining the length of a curve described by a function. When you have a function in the form of \( y = f(x) \), and you're interested in the length of the curve between two points \( x = a \) and \( x = b \), you use the following formula:

• \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]

This formula essentially computes the continuous sum of infinitesimally small line segments along the curve. Each segment is a combination of a small change in \( x \) and the change in \( y \) given by the derivative \( \frac{dy}{dx} \).
This ensures that the pieces add up to reflect the true path of the curve across the given interval.

Calculus Problem

When tackling a calculus problem like finding the arc length of a curve, it's essential to first understand the function and the interval you are working with. For the problem at hand, the function is \( y = x^{3/2} \) and the interval is from \( x = 0 \) to \( x = 4 \).

• The task involves applying calculus techniques, such as differentiation and integration, to find an exact measurement for the length.
• This type of problem is typical in calculus where you transition from understanding a theoretical concept to applying mathematical processes.

Breaking it down step by step helps in seeing how each part of the calculus process interconnects, leading you to find the arc length efficiently.

Integral Calculation

Integral calculation is fundamental for determining the arc length. Once you substitute the derivative into the arc length formula, it requires evaluation through integration.

• The integral in this case is \[ \int_{0}^{4} \sqrt{1 + \frac{9}{4}x} \, dx \], which needs to be solved to provide the length of the curve.
• To tackle this, a substitution technique is often used to simplify the integrand, making it easier to calculate.

Substituting \( u = 1 + \frac{9}{4}x \) transforms the integral to a simpler form. Calculating this integral involves evaluating the function from the lower bound to the upper bound of the interval, which provides the numerical result for the arc length.

Function Derivatives

Understanding function derivatives is necessary when finding the arc length. The derivative \( \frac{dy}{dx} \) provides the rate of change of the function at any point along its curve.

• For the function \( y = x^{3/2} \), use the power rule to differentiate, resulting in \( \frac{dy}{dx} = \frac{3}{2}x^{1/2} \).
• This derivative tells us how steep the curve is at any point \( x \) from 0 to 4.

The derivative is then squared and added to 1 inside the square root in the arc length formula. This step is crucial, providing the change in the curve's height that accumulates over the interval. Understanding derivatives means you'll handle other similar calculus problems more effectively.