Problem 3 - Thomas Calculus

Question

In Exercises \(1 - 8 :\) 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. $$x y = 1 , \quad 1 \leq y \leq 2 ; \quad y -axis$$

Step-by-Step Solution

Answer

Set up and evaluate \( 2\pi \int_{1}^{2} \frac{1}{y} \sqrt{1 + \frac{1}{y^4}} \, dy \) to find the surface area numerically.

Step 1: Understand the Problem

We need to find the surface area of a curve given by the equation \( xy = 1 \), as it revolves around the \( y \)-axis, between \( y = 1 \) and \( y = 2 \).

Step 2: Express x as a Function of y

Start by expressing \( x \) in terms of \( y \) using the curve equation. Since \( xy = 1 \), solving for \( x \) gives us \( x = \frac{1}{y} \).

Step 3: Formula for Surface Area of Revolution

To find the surface area \( S \), we use the formula: \[ S = 2 \pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy \] where \( x \) is a function of \( y \).

Step 4: Derivative of x with Respect to y

Compute the derivative \( \frac{dx}{dy} \) of \( x = \frac{1}{y} \). The derivative is \( \frac{dx}{dy} = -\frac{1}{y^2} \).

Step 5: Determine the Integral

Substitute \( x = \frac{1}{y} \) and \( \frac{dx}{dy} = -\frac{1}{y^2} \) into the surface area formula. Evaluate:\[ S = 2 \pi \int_{1}^{2} \frac{1}{y} \sqrt{1 + \left( -\frac{1}{y^2} \right)^2} \, dy \]This simplifies to:\[ S = 2 \pi \int_{1}^{2} \frac{1}{y} \sqrt{1 + \frac{1}{y^4}} \, dy \]

Step 6: Graphing the Curve and Surface

Graph the curve \( xy = 1 \), which is a hyperbola. For the surface, visualize it as a hyperboloid obtained by rotating this hyperbola around the \( y \)-axis.

Step 7: Evaluate the Integral Numerically

Using a calculator or software, evaluate the integral numerically:\[ S = 2 \pi \int_{1}^{2} \frac{1}{y} \sqrt{1 + \frac{1}{y^4}} \, dy \]This gives the surface area. Numerical tools like WolframAlpha or a graphing calculator can be used to get a numerical solution.

Key Concepts

Integral Calculus

Integral Calculus is a branch of mathematics focused on accumulation and area calculation. It involves finding integrals, which can tell us about total changes, areas under curves, and several other applications.
Understanding integrals and how they work is crucial for calculating surface areas when curves revolve around an axis. Integrals allow us to accumulate an infinite number of infinitesimal pieces to see the whole picture. This makes it possible to calculate complex areas that would be hard to measure directly.

• Definite Integrals: These are used to calculate the area under a curve within a specific interval.
• Indefinite Integrals: These serve to find a general formula for area calculation, often including a constant.

When computing the surface area of a revolution, as in our exercise, we are using definite integrals to accumulate the infinitesimally small 'rings' created by rotation.

Hyperboloid

A hyperboloid is a specific 3D shape that resembles either a stretched or a squashed cylinder. It can be categorized as one of two types: one-sheeted or two-sheeted, depending on its properties and appearance.
The exercise involves rotating the hyperbola given by the equation \(xy = 1\) around the \(y\)-axis. This rotation creates a hyperbolic shape known as a hyperboloid. Luckily, understanding what a hyperboloid looks like helps us visualize the area that we want to calculate.

• One-Sheeted Hyperboloid: This resembles an hourglass shape, often seen in cooling towers of power plants.
• Two-Sheeted Hyperboloid: Appears more like two separate dishes stacked together, stemming from a different type of rotation.

The rotation of the curve in the problem gives rise to a one-sheeted hyperboloid.

Numerical Integration

Numerical integration is a powerful tool when exact solutions to integrals are difficult or impossible to find analytically. It uses numerical approximations to estimate the value of an integral.
This method is particularly helpful in cases like our exercise, where calculating the integral of the surface area formula directly might not be straightforward. Numerical integration allows us to compute approximations effectively using software or calculators.

• Simpson’s Rule: This technique uses parabolic approximations to estimate integrals. It's especially useful for smoother curves.
• Trapezoidal Rule: This approach uses linear approximations to get an estimation by creating trapezoids under the curve.

With numerical integration, even complex or non-standard formulas can be evaluated to get a practical answer.

Surface Area Formula

The surface area formula for revolution is crucial in this context, allowing us to calculate the area of a surface generated by rotating a curve around an axis. This formula gives us a way to measure what might otherwise be an inaccessible area.
The formula used in this exercise for the surface area \(S\) is:\[ S = 2 \pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy \]Here's the breakdown of the components:

• \(x\): Represents the function being rotated, expressed in terms of another variable (here it's \(y\)).
• \(\frac{dx}{dy}\): The derivative represents the rate of change of \(x\) with respect to \(y\).

This formula helps bundle the results of small rotations into an entire surface area. It's essential for converting between the geometric shape we visualize and the calculable quantity we derive from it.