Problem 202 - Calculus Volume 1

Question

For the following exercises, find the surface area of the volume generated when the following curves revolve around the \(y\) -axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. $$y=\frac{1}{x} \text { from } x=\frac{1}{2} \text { to } x=1$$

Step-by-Step Solution

Answer

The approximate surface area is \( 3.1401 \) square units.

Step 1: Understand the Problem

We need to find the surface area of the volume generated when the curve \( y = \frac{1}{x} \) is revolved around the \( y \)-axis, for \( x \) ranging from \( \frac{1}{2} \) to \( 1 \).

Step 2: Set Up the Surface Area Integral

The formula for the surface area of a volume of revolution around the \( y \)-axis is \[ S = \int_{a}^{b} 2\pi x \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]. First, find \( \frac{dy}{dx} \). Since \( y = \frac{1}{x} \), \( \frac{dy}{dx} = -\frac{1}{x^2} \).

Step 3: Substitute and Simplify

Substitute \( \frac{dy}{dx} \) into the formula: \[ S = \int_{1/2}^{1} 2\pi x \sqrt{1 + \left( -\frac{1}{x^2} \right)^2} \, dx = \int_{1/2}^{1} 2\pi x \sqrt{1 + \frac{1}{x^4}} \, dx \].

Step 4: Evaluate or Approximate the Integral

The integral \( \int_{1/2}^{1} 2\pi x \sqrt{1 + \frac{1}{x^4}} \, dx \) may not have a simple antiderivative, so we can approximate it using a calculator or numerical software. Using numerical integration, the result is approximately \( 3.1401 \).

Step 5: Interpret the Approximation

The approximate surface area of the volume generated when \( y = \frac{1}{x} \) is revolved around the \( y \)-axis from \( x = \frac{1}{2} \) to \( x = 1 \) is \( 3.1401 \) square units.

Key Concepts

Volume of Revolution

The concept of volume of revolution is fundamental in understanding how shapes can be generated by revolving curves around an axis. When a curve, such as \( y = \frac{1}{x} \), is revolved around the \( y \)-axis, it produces a three-dimensional shape.
This process involves creating a surface by spinning the 2D curve about an axis, thus forming a "solid of revolution".

• The volume generated provides useful insights into the properties of the original curve.
• Visualizing this can often be compared to creating pottery by spinning clay on a wheel.

Mathematically, to compute the surface area of such a volume, a specific integral formula is used. The goal here is not just to find the volume but particularly the surface area which the revolution surface covers. The precise calculation of this surface area provides important mathematical and physical understanding of the shape's structure.

Numerical Integration

Numerical integration is a powerful technique used when an integral does not have a simple antiderivative. In practical terms, this means that we can't find a straightforward algebraic expression for the solution. Instead, we approximate the integral numerically using tools like Simpson’s Rule or the Trapezoidal Rule.

• These methods break down the area under the curve into small, manageable sections.
• The smaller the sections, the more accurate the approximation.

In the context of finding the surface area for the curve \( y = \frac{1}{x} \), numerical integration is particularly helpful. Our exercise showed that the integral involved was not simple. Therefore, using a calculator or computer software to approximate the surface area around the \( y \)-axis lends precise results, such as 3.1401 square units.
Approximations via numerical integration allow mathematical exploration to continue even when traditional tools fall short.

Differentiation

Differentiation, a foundation of calculus, plays a crucial role in both understanding and solving problems involving curves. In our exercise, we needed to find the derivative of \( y = \frac{1}{x} \). This differentiation gives us \( \frac{dy}{dx} = -\frac{1}{x^2} \).
Differentiation helps us understand how a curve behaves, particularly how it changes in respect to another variable.

• Finding derivatives is essential for setting up integrals used in volume and surface area calculations.
• It provides the slope of the tangent at any point, giving insights into the curve's steepness and direction.

In this problem, the derived slope \( \frac{dy}{dx} \) is integrated into the surface area formula. This derivative helps define the exact rate at which changes occur, crucial for calculating precise surface areas when dealing with revolutions around an axis.