Problem 8 - Thomas Calculus in SI Units

Question

a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$y=\int_{1}^{x} \sqrt{t^{2}-1} d t, \quad 1 \leq x \leq \sqrt{5} ; \quad x \text { -axis }$$

Step-by-Step Solution

Answer

Set up and simplify the integral. Evaluate it numerically for the surface area.

Step 1: Define the Curve

The given curve is described by the integral function \( y = \int_{1}^{x} \sqrt{t^{2}-1} \, dt \). This represents the area under the curve \( \sqrt{t^{2}-1} \) from 1 to \( x \). We will use this to understand what function \( y \) plots as.

Step 2: Set Up the Surface Area Integral

To find the surface area of the solid formed by revolving the curve about the x-axis, use the formula for the surface area of a solid of revolution: \[ A = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]Find \( \frac{dy}{dx} = \sqrt{x^2 - 1} \) and substitute \( y \) and \( \frac{dy}{dx} \) to set up the integral:\[ A = 2\pi \int_{1}^{\sqrt{5}} \left( \int_{1}^{x} \sqrt{t^{2}-1} \, dt \right) \sqrt{1 + (x^2 - 1)} \, dx \]

Step 3: Simplify the Integral Expression

Simplify the expression \( \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \) by substituting \( \frac{dy}{dx} = \sqrt{x^2 - 1} \). This gives:\[ \sqrt{1 + (x^2 - 1)} = \sqrt{x^2} = x \]The integral becomes:\[ A = 2\pi \int_{1}^{\sqrt{5}} \left( \int_{1}^{x} \sqrt{t^{2}-1} \, dt \right) \cdot x \, dx \]

Step 4: Graph the Curve and Surface

Use a graphing utility to plot the function \( y = \int_{1}^{x} \sqrt{t^2-1} \, dt \) over the interval \([1, \sqrt{5}]\). To graph the surface of revolution, visualize the curve rotating around the x-axis. This will create a 3D surface, often represented as a series of concentric rings or a shaded surface.

Step 5: Evaluate the Integral Numerically

Use a calculator or computer algebra system (CAS) to numerically evaluate the integral:\[ A = 2\pi \int_{1}^{\sqrt{5}} \left( \int_{1}^{x} \sqrt{t^{2}-1} \, dt \right) \cdot x \, dx \]This will provide the approximate surface area of the revolution.

Key Concepts

Surface of Revolution

A surface of revolution is formed when a curve is revolved around a given axis, creating a 3D shape. The most common axis of revolution is the x-axis, but it can also be the y-axis or any other line in the plane. In this exercise, the curve described by the integral function \( y = \int_{1}^{x} \sqrt{t^{2}-1} \, dt \) is revolved around the x-axis.
This action generates what is known as a "solid of revolution." To find the surface area of this solid, we use a specific formula that incorporates the curve's original equation.

Here's how it works: the formula for the surface area \( A \) of the solid created by revolving a curve \( y = f(x) \) from \( x = a \) to \( x = b \) around the x-axis is:
\[ A = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
This formula combines the distance the curve moves as it rotates (represented by \( 2\pi y \)), with an adjustment factor \( \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \) which accounts for changes in direction of the curve along x.
This results in the total area of the surface created by the revolution.

Numerical Integration

Numerical integration is a technique used to approximate the value of integrals, which can be especially useful when an integral does not have a simple or closed-form solution. In this exercise, the integral representing the surface area of a revolution is evaluated using numerical methods.

The integral set up to calculate the surface area is:
\[ A = 2\pi \int_{1}^{\sqrt{5}} \left( \int_{1}^{x} \sqrt{t^{2}-1} \, dt \right) \cdot x \, dx \]

This expression involves an inner integral \( \int_{1}^{x} \sqrt{t^{2}-1} \, dt \), complicating the evaluation. A numerical approach helps us find a definite value for this integral from 1 to \( \sqrt{5} \).

Common numerical integration techniques include methods like:

• Trapezoidal rule
• Simpson's rule
• Numerical algorithms found in calculators or computer algebra systems (CAS)

These methods allow students to find approximate values for integrals that are either too complex for analytical methods or where exact values are not required for practical purposes.

Graphing Utilities

Graphing utilities are tools that help visualize mathematical equations, making it easier to understand complex concepts. In the context of this exercise, graphing utilities play a crucial role in visualizing both the curve and the surface of revolution.

To begin, use a graphing calculator or software like Desmos, GeoGebra, or graphing capabilities within a CAS to plot the function \( y = \int_{1}^{x} \sqrt{t^2-1} \, dt \). By graphing the curve over the specified interval \([1, \sqrt{5}]\), students can observe its behavior and shape in a 2D plane.

For the surface of revolution, some advanced graphing tools allow you to simulate the 3D surface created by rotating the curve around the x-axis. These visualizations can resemble a series of concentric rings, or a full shaded surface, providing a clear view of the solid generated.

Graphing utilities also assist in verifying the setup for integrals and checking calculations, giving visual confirmation of the logical steps involved in solving problems related to calculus and surface area integrals.