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Volumes of Revolution

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Volumes of Revolution - Complete Interactive Lesson

Part 1: Disk Method

Volumes of Revolution

Part 1 of 7 — The Disk Method

Topic Overview

Part	Topic
1	Disk method
2	Washer method
3	Rotation about other lines
4	Cross-sectional volumes
5	Disk & washer in $y$
6	AP-style workshop
7	Comprehensive assessment

The Disk Method

When you rotate a single curve around an axis, each cross-section is a disk (circle):

$\boxed{V = \pi\int_a^b [R(x)]^2\,dx}$

Key Fact: $R(x)$ is the distance from the curve to the axis of rotation. For rotation about the $x$ -axis, $R(x) = f(x)$ .

Step-by-Step

Step	Action
1	Identify the axis of rotation
2	Find $R(x) =$ distance from curve to axis
3	Set up $\pi\int_a^b R^2\,dx$

Worked Example

Rotate $y = \sqrt{x}$ about the $x$ -axis from $x=0$ to .

$R(x) = \sqrt{x}$ .

$V = \pi\int_0^4(\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx = \pi\left[\frac{x^2}{2}\right]_0^4 = \boxed{8\pi}$

AP Tip: The most common mistake is forgetting to square $R(x)$ or forgetting $\pi$ . Always write the formula first.

Practice — Disk Method 🎯

Identify the setup. 🔍

Compute. ✍️

Key Takeaways — Part 1

Disk method: $V = \pi\int_a^b R^2\,dx$
= distance from curve to axis of rotation

Part 2: Washer Method

Volumes of Revolution

Part 2 of 7 — The Washer Method

When There’s a Hole

When the region does not touch the axis of rotation, each cross-section is a washer (ring):

$\boxed{V = \pi\int_a^b\left([R(x)]^2 - [r(x)]^2\right)dx}$

Part 3: Rotation About Other Axes

Volumes of Revolution

Part 3 of 7 — Rotation About Other Lines

Adjusting Radii for Non-Standard Axes

$\boxed{R = |f(x) - k|, \quad r = |g(x) - k|}$

Part 4: Cross-Sectional Volumes

Volumes of Revolution

Part 4 of 7 — Cross-Sectional Volumes

Known Cross-Sections (Not Revolution!)

Instead of rotating, cross-sections of known shapes are stacked along an axis:

$\boxed{V = \int_a^b A(x)\,dx}$

Part 5: Disk/Washer in y

Volumes of Revolution

Part 5 of 7 — Disk & Washer in $y$

Rotating About the $y$ -axis

When rotating about the $y$ -axis, express curves as functions of $y$ and integrate in $dy$ :

Part 6: AP-Style Workshop

Volumes of Revolution

Part 6 of 7 — AP-Style Workshop

Typical AP FRQ Structure

The AP exam usually defines a region $R$ and asks:

Part	Prompt	Method
(a)	Find area of $R$	$\int(\text{top}-\text{bottom})$

Part 7: Comprehensive Assessment

Volumes of Revolution

Part 7 of 7 — Comprehensive Assessment

Complete Formula Reference

Method	Formula	When to Use
Disk	$\pi\int R^2\,dx$	Region touches axis
Washer	$π \int (R^{2} - r^{2})$

Don’t forget to square

R

and include

\pi

Works when the region touches the axis (no hole)

Term	Meaning
$R(x)$	Outer radius (curve farther from axis)
$r(x)$	Inner radius (curve closer to axis)

Key Fact: NEVER subtract radii first! It’s $R^2 - r^2$ , NOT $(R-r)^2$ . This is the most common washer mistake.

Region between $y = x$ and $y = x^2$ on $[0,1]$ , rotated about $x$ -axis.

Outer: $R = x$ (farther from axis). Inner: $r = x^2$ (closer).

$V = \pi\int_0^1(x^2 - x^4)\,dx = \pi\left[\frac{x^3}{3}-\frac{x^5}{5}\right]_0^1 = \pi\left(\frac{1}{3}-\frac{1}{5}\right) = \boxed{\frac{2\pi}{15}}$

Feature	Disk	Washer
Region touches axis?	Yes	No
Cross-section	Full circle	Ring (annulus)
Formula	$\pi\int R^2$	$\pi\int(R^2-r^2)$
Inner radius	$r = 0$	$r \neq 0$

Practice — Washer Method 🎯

Classify each setup. 🔍

Calculate. ✍️

Key Takeaways — Part 2

Washer method: $V = \pi\int(R^2-r^2)\,dx$
$R$ = outer radius, $r$ = inner radius (from axis)
NEVER square $(R-r)$ — always $R^2 - r^2$
Disk method is a washer with $r = 0$

where $y = k$ is the axis of rotation.

Axis	Outer Radius	Inner Radius
$x$ -axis ( $y=0$ )	$f(x)$	$g(x)$
$y = k$ below region	$f(x)-k$	$g(x)-k$
$y = k$ above region	$k - g(x)$	$k - f(x)$
$y$ -axis ( $x=0$ )	Use $dy$	Use $dy$

Key Fact: When the axis is above the region, the farther curve becomes the outer radius and the closer curve becomes the inner radius. Which is "outer" can flip!

Worked Example 1 — Axis Below

Rotate region between $y = x^2$ and $y = 1$ about $y = -1$ on $[-1,1]$ .

$R = 1-(-1) = 2$ (outer: $y=1$ is farther). $r = x^2-(-1) = x^2+1$ (inner: $y=x^2$ is closer).

Wait — which is farther from $y = -1$ ? At $x = 0$ : $y=1$ gives distance $2$ , $y=0$ gives distance $1$ . So $y=1$ is outer.

$V = \pi\int_{-1}^1[4-(x^2+1)^2]\,dx = 2\pi\int_0^1[4-x^4-2x^2-1]\,dx = 2\pi\int_0^1(3-2x^2-x^4)\,dx$

$= 2\pi\left[3x-\frac{2x^3}{3}-\frac{x^5}{5}\right]_0^1 = 2\pi\left(3-\frac{2}{3}-\frac{1}{5}\right) = 2\pi \cdot \frac{32}{15} = \boxed{\frac{64\pi}{15}}$

Worked Example 2 — Axis Above

Rotate region between $y = x^2$ and $y = 1$ about $y = 3$ on $[-1,1]$ .

$R = 3-x^2$ (outer: $y=x^2$ is farther from $y=3$ ). $r = 3-1 = 2$ (inner: $y=1$ is closer).

$V = \pi\int_{-1}^1[(3-x^2)^2-4]\,dx$

Practice — Non-Standard Axes 🎯

Identify the radii. 🔍

Calculate. ✍️

Key Takeaways — Part 3

Radius = distance from curve to axis: $|f(x) - k|$
Axis below: outer = farther curve ( $f(x)-k$ )
Axis above: outer = closer-to-ground curve ( $k - g(x)$ is larger)
Always test: which curve is farther from the axis?

where $A(x)$ is the area of the cross-section at position $x$ .

Key Fact: No $\pi$ in the formula (unless the cross-section is a semicircle). The $\pi$ in disk/washer comes from circular cross-sections.

Cross-Section Area Formulas

If the side length is $s = f(x) - g(x)$ :

Shape	Area Formula
Square	$A = s^2$
Semicircle (diameter $= s$ )	$A = \frac{\pi s^2}{8}$
Equilateral triangle	$A = \frac{\sqrt{3}}{4}s^2$
Isosceles right triangle (leg $= s$ )	$A = \frac{1}{2}s^2$
Isosceles right triangle (hyp $= s$ )	$A = \frac{1}{4}s^2$

Base: region between $y = \sqrt{x}$ and $y = 0$ on $[0,4]$ . Cross-sections $\perp$ to $x$ -axis are squares.

Side $= \sqrt{x} - 0 = \sqrt{x}$ . Area $= (\sqrt{x})^2 = x$ .

$V = \int_0^4 x\,dx = \left[\frac{x^2}{2}\right]_0^4 = \boxed{8}$

AP Tip: Cross-section volume problems are one of the most frequently tested FRQ topics. Practice identifying which formula to use from the shape name.

Practice — Cross-Sections 🎯

Match the shape to the formula. 🔍

Calculate. ✍️

Key Takeaways — Part 4

Cross-section volume: $V = \int A(x)\,dx$ — no automatic $\pi$
Find the side length from the base region
Memorize the 5 common cross-section area formulas
Very common on AP FRQ problems

\boxed{V = \pi\int_c^d [R(y)]^2\,dy \quad \text{(disk)}}

$\boxed{V = \pi\int_c^d\left([R(y)]^2-[r(y)]^2\right)dy \quad \text{(washer)}}$

Rotate about...	Integrate in...	Radii are functions of...
$x$ -axis or $y = k$	$dx$	$x$
$y$ -axis or $x = k$	$dy$	$y$

$y = x^2$ from $y=0$ to $y=4$ , rotated about the $y$ -axis.

Solve for $x$ : $x = \sqrt{y}$ . Radius $R(y) = \sqrt{y}$ .

$V = \pi\int_0^4(\sqrt{y})^2\,dy = \pi\int_0^4 y\,dy = \pi\left[\frac{y^2}{2}\right]_0^4 = \boxed{8\pi}$

Washer in $y$ Example

Region between $x = y$ and $x = y^2$ on $[0,1]$ , rotated about the $y$ -axis.

Outer: $R = y$ (farther from $y$ -axis). Inner: $r = y^2$ .

$V = \pi\int_0^1(y^2-y^4)\,dy = \pi\left[\frac{y^3}{3}-\frac{y^5}{5}\right]_0^1 = \pi\left(\frac{1}{3}-\frac{1}{5}\right) = \boxed{\frac{2\pi}{15}}$

Practice — $y$ -Axis Rotation 🎯

Verify your reasoning. 🔍

Key Takeaways — Part 5

For $y$ -axis rotation: use $dy$ , express $x$ as function of $y$
For $x = k$ rotation: radii measured horizontally from $x=k$
Same disk/washer formulas — just swap the variable roles
Limits are $y$ -values when integrating in $dy$

$R$ is bounded by $y = \sqrt{x}$ , $y = 0$ , $x = 4$ .

(a) Area: $A = \int_0^4\sqrt{x}\,dx = \frac{2}{3}(4^{3/2}) = \frac{16}{3}$

(b) Cross-sections are squares:

Side $= \sqrt{x}$ . $V = \int_0^4 x\,dx = 8$ .

(c) Rotate about $x$ -axis:

$V = \pi\int_0^4 x\,dx = 8\pi$ .

(d) Rotate about $y = 3$ . Write but do not evaluate:

$R = 3$ (from $y=0$ ), $r = 3-\sqrt{x}$ .

$V = \pi\int_0^4\left[9-(3-\sqrt{x})^2\right]dx$

AP Tip: For "write but do not evaluate," show the integral with correct limits and integrand. Do NOT expand or simplify — this earns full credit and avoids algebra errors.

AP Practice 🎯

AP decision-making. 🔍

AP Challenge. ✍️

Key Takeaways — Part 6

AP FRQs combine area, cross-section, and revolution in one problem
"Write but do not evaluate" = set up only, do not simplify
Know cross-section formulas by heart
Check: does region touch the axis? (disk vs washer)

π \int (R^{2} - r^{2}) d x

Mistake	Correction
$(R-r)^2$ instead of $R^2-r^2$	Expand: they are NOT equal
Forgetting $\pi$	Revolution always has $\pi$ ; cross-section may not
Wrong axis → wrong radii	$R =
Using $dx$ for $y$ -axis rotation	Match variable to perpendicular direction
Wrong cross-section formula	Memorize all 5 shapes
Not showing setup on FRQ	Write integral before evaluating

Quiz — Methods 🎯

Quiz — Cross-Sections & Setup 🎯

Final classification. 🔍

Final Challenge. ✍️

Volumes of Revolution — Complete!

You’ve mastered:

Part	Topic
1	Disk method
2	Washer method
3	Rotation about other lines
4	Cross-sectional volumes
5	Disk & washer in $y$
6	AP-style workshop
7	Comprehensive assessment

You’re ready for AP-level volume problems!

Volumes of Revolution

Worked Example

Disk vs Washer

Key Takeaways — Part 2

Quick Reference

Worked Example 1 — Axis Below

Worked Example 2 — Axis Above

Key Takeaways — Part 3

Cross-Section Area Formulas

Worked Example

Key Takeaways — Part 4

When to Use $dy$

Worked Example

Washer in $y$ Example

Key Takeaways — Part 5

Worked AP Problem

Key Takeaways — Part 6

Top AP Mistakes

Volumes of Revolution — Complete!

Volumes of Revolution

Volumes of Revolution - Complete Interactive Lesson

Part 1: Disk Method

Volumes of Revolution

Topic Overview

The Disk Method

Step-by-Step

Worked Example

Key Takeaways — Part 1

Part 2: Washer Method

Volumes of Revolution

When There’s a Hole

Part 3: Rotation About Other Axes

Volumes of Revolution

Adjusting Radii for Non-Standard Axes

Part 4: Cross-Sectional Volumes

Volumes of Revolution

Known Cross-Sections (Not Revolution!)

Part 5: Disk/Washer in y

Volumes of Revolution

Rotating About the yyy-axis

Part 6: AP-Style Workshop

Volumes of Revolution

Typical AP FRQ Structure

Part 7: Comprehensive Assessment

Volumes of Revolution

Complete Formula Reference

Worked Example

Disk vs Washer

Key Takeaways — Part 2

Quick Reference

Worked Example 1 — Axis Below

Worked Example 2 — Axis Above

Key Takeaways — Part 3

Cross-Section Area Formulas

Worked Example

Key Takeaways — Part 4

When to Use dydydy

Worked Example

Washer in yyy Example

Key Takeaways — Part 5

Worked AP Problem

Key Takeaways — Part 6

Top AP Mistakes

Volumes of Revolution — Complete!

Rotating About the $y$ -axis

When to Use $dy$

Washer in $y$ Example