Problem 6 - Thomas Calculus

Question

In Exercises \(1-8 :\) $$\begin{array}{l}{\text { a. Set up an integral for the area of the surface generated by }} \\ {\text { revolving the given curve about the indicated axis. }} \\ {\text { b. Graph the curve to see what it looks like. If you can, graph }} \\ {\text { the surface too. }} \\ {\text { c. Use your utility's integral evaluator to find the surface's area }} \\ {\text { numerically. }}\end{array}$$ $$ y+2 \sqrt{y}=x, \quad 1 \leq y \leq 2 ; \quad y\text-axis $$

Step-by-Step Solution

Answer

The surface area is found using the integral \( S = 2\pi \int_{1}^{2} (y + 2\sqrt{y}) \sqrt{2 + 2\frac{1}{\sqrt{y}} + \frac{1}{y}} \dy \).

Step 1: Parametrize the Curve

First, express the given equation in terms of one variable. The equation is given as \( y + 2\sqrt{y} = x \). Thus, the parametrized function is \( x(y) = y + 2\sqrt{y} \).

Step 2: Compute the Derivative

Find the derivative of \( x(y) \) with respect to \( y \). This is \( x'(y) = 1 + \frac{1}{\sqrt{y}} \).

Step 3: Set up the Integral for Surface Area

The formula for the surface area of a revolution about the y-axis is \[ S = 2\pi \int_{a}^{b} x(y) \sqrt{1 + (x'(y))^2} \, dy \]. Substitute \( x(y) = y + 2\sqrt{y} \) and \( x'(y) = 1 + \frac{1}{\sqrt{y}} \).

Step 4: Simplify the Expression

Simplify \( \sqrt{1 + (x'(y))^2} = \sqrt{1 + \left(1 + \frac{1}{\sqrt{y}}\right)^2} \). This equals \( \sqrt{1 + 1 + 2\frac{1}{\sqrt{y}} + \frac{1}{y}} = \sqrt{2 + 2\frac{1}{\sqrt{y}} + \frac{1}{y}} \).

Step 5: Set up and Evaluate the Integral

The integral becomes \[ S = 2\pi \int_{1}^{2} (y + 2\sqrt{y}) \sqrt{2 + 2\frac{1}{\sqrt{y}} + \frac{1}{y}} \dy \]. Use your calculator or software to evaluate this integral numerically.

Step 6: Graph the Curve and Surface

Plot the curve \( y + 2\sqrt{y} = x \) for \( 1 \leq y \leq 2 \). If possible, also plot the surface obtained by revolving the curve about the y-axis using graphing software or a calculator.

Key Concepts

Integral Setup

When calculating the surface area of a surface formed by revolving a curve about an axis, setting up the integral correctly is crucial. In our exercise, we are working with the curve given by the equation \( y + 2\sqrt{y} = x \). The task is to find the surface area when this curve is revolved around the y-axis.
To do this, we use the surface area formula for revolution about the y-axis:

• \( S = 2\pi \int_{a}^{b} x(y) \sqrt{1 + (x'(y))^2} \, dy \)

Here, \( x(y) \) is our original equation written in terms of \( y \), and \( x'(y) \) is its derivative. This setup will help us correctly measure 'slices' of the surface as it rotates around the y-axis, allowing us to accumulate the total area.
Substitute the expressions for \( x(y) = y + 2\sqrt{y} \) and \( x'(y) = 1 + \frac{1}{\sqrt{y}} \) into the formula. This gives us the integral:

• \( S = 2\pi \int_{1}^{2} (y + 2\sqrt{y}) \sqrt{2 + 2\frac{1}{\sqrt{y}} + \frac{1}{y}} \, dy \)

Now, with the integral set up, we can proceed to solve it using numerical integration.

Parametrization

Parametrization is a method used to express the equations governing your curve in a single variable. It simplifies the calculation process, especially when dealing with derivatives and integrals. In this exercise, we need to express the given equation \( y + 2\sqrt{y} = x \) in terms of \( y \). The function \( x(y) \) depends only on \( y \).
Parametrization helps to handle more complex problems by breaking down expressions into manageable elements. Here, our goal is to have a single equation \( x(y) \) that expresses the relationship fully in terms of \( y \).
This process allows us to easily compute derivatives and setup integrals. By finding \( x'(y) = 1 + \frac{1}{\sqrt{y}} \), we handle changes to the x-value as \( y \) changes, which is crucial for accurately determining the surface area in the following steps.

Numerical Integration

Numerical integration becomes necessary when a given integral does not have a simple analytical solution. In this exercise, once we have set up the integral for the surface area, we need numerical methods to evaluate it.
Using software, calculators, or integration tools, we can approximate the value of the area \( S = 2\pi \int_{1}^{2} (y + 2\sqrt{y}) \sqrt{2 + 2\frac{1}{\sqrt{y}} + \frac{1}{y}} \, dy \). This process involves:

• Choosing a method such as Simpson's Rule, Trapezoidal Rule, or using built-in functions in calculator software.
• Inputting the bounds of integration, in our case from \( y = 1 \) to \( y = 2 \).
• Running the calculation to find an approximate area value.

Numerical integration provides us with a practical means of determining the integral's value, which might not be achievable by hand, especially if the expression inside the integral complicates the procedure.

Graphing Curves

Graphing curves in mathematics helps visualize how a particular function behaves within a specific interval. In our exercise, graphing the curve defined by \( y + 2\sqrt{y} = x \) from \( 1 \leq y \leq 2 \) gives us a visual insight into the function's shape.
Utilizing graphing tools or software, we plot this equation to visualize not only the original curve but also how it will appear when revolved around the y-axis. This visual understanding is key in conceptualizing the surface of revolution formed by the curve.
When attempting to graph such a curve:

• Use a graphing calculator or software, inputting the equation with the given range.
• Observe the plot to understand the direction and style of the curve.
• If possible, try using 3D graphing tools to view the surface formed by revolving the curve around the y-axis.

Topographic visualizations like these significantly enhance comprehension by providing tangible insights into abstract mathematical concepts.