Volume 3: The Calculus of Life

Workbook 27.3: The Shell Method

Directives for the Forester:

1. Identify the Layers: Use shells when rotating around the Y-axis with a $dx$ integral.
2. Set the Radius ($r$): The radius is usually the distance $x$.
3. Set the Height ($h$): The height is the function $f(x)$.
4. The Sum: Volume = $2\pi \int [x \cdot f(x)] dx$.

Part I: Basic Shells (Rotating around Y)

Find the volume of the solid generated by rotating the region under $f(x)$ about the Y-axis.

$f(x) = x$ from $x=0$ to $x=2$.
Note: This creates a Cone (opening up)!

$r = x, h = x$
$V = 2\pi \int_{0}^{2} (x \cdot x) dx = 2\pi \int_{0}^{2} x^2 dx = ...$

$f(x) = x^3$ from $x=0$ to $x=1$.

$r = x, h = x^3$
$V = 2\pi \int_{0}^{1} x^4 dx = ...$

$f(x) = \sin(x^2)$ from $x=0$ to $x=\sqrt{\pi}$.

$r = x, h = \sin(x^2)$
$V = 2\pi \int_{0}^{\sqrt{\pi}} x \sin(x^2) dx = ...$
Hint: Use u-substitution! $u = x^2$.
The Logic Check:

In the first problem (the Cone), use the Disk method to find the same volume. (Rotate $y=x$ around the Y-axis). Which method was easier for you? Why does God provide "Alternate Paths" to the same solution?

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Part II: Hollow Shells (Between Curves)

Rotate the region between $y = 4x - x^2$ and $y = 0$ around the Y-axis.

$r = x, h = (4x - x^2)$
$V = 2\pi \int_{0}^{4} x(4x - x^2) dx = ...$

Part III: The Challenge (The Shifted Shell)

The Off-Center Onion

Rotate the region under $y = x^2$ from $x=0$ to $x=2$ around the line **$x = -1$**.
Task: Find the new radius ($r = x + 1$). Set up and solve the integral.

$V = 2\pi \int_{0}^{2} (x+1)(x^2) dx$
$V = 2\pi \int (x^3 + x^2) dx = ...$

Part IV: Transmission (The Echad Extension)

Teacher Log: The Unrolled Tube

Objective: Explain the Shell Method to a younger student using a paper towel roll.

The Activity:
1. Show the roll. "It's a cylinder."
2. Cut it down the side and unroll it.
3. "Now it's a rectangle! The width is the 'Around' part ($2\pi r$). The height is the 'Up' part ($h$)."

The Lesson: "Math can unroll our complicated life into simple rectangles so we can add them up easily."


Response: ___________________________________________________________

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