🎯⭐ INTERACTIVE LESSON

Volumes of Revolution

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Volumes of Revolution - Complete Interactive Lesson

Part 1: Disk Method

Volumes of Revolution

Part 1 of 7 — Disk Method

Rotating Around the x-axis

When rotating $y = f(x)$ around the $x$ -axis, each cross-section is a disk with radius $f(x)$ :

$V = \pi\int_a^b [f(x)]^2\,dx$

Worked Example

Find the volume when $y = \sqrt{x}$ from $x = 0$ to is rotated about the -axis.

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

Disk Method 🎯

Key Takeaways — Part 1

Disk method: $V = \pi\int_a^b [R(x)]^2\,dx$

Part 2: Washer Method

Volumes of Revolution

Part 2 of 7 — Washer Method

When There's a Hole

Rotating a region between two curves creates a washer (disk with a hole):

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

Part 3: Rotation About Other Axes

Volumes of Revolution

Part 3 of 7 — Rotation About Other Axes

Rotating About $y = k$ or $x = k$

When rotating about a line other than the $x$ -axis, adjust the radii:

Rotation about $y = k$ (horizontal line):

Part 4: Cross-Sectional Volumes

Volumes of Revolution

Part 4 of 7 — Cross-Sectional Volumes

Known Cross-Sections

Instead of rotating, we can have cross-sections that are squares, semicircles, equilateral triangles, etc.

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

where is the area of the cross-section at position .

Part 5: Disk/Washer in y

Volumes of Revolution

Part 5 of 7 — Disk/Washer in $y$

Rotating About the $y$ -axis Using $dy$

$V = \pi\int_c^d [R(y)]^2\,dy$

Part 6: AP-Style Workshop

Volumes of Revolution

Part 6 of 7 — AP-Style Workshop

AP-Style Volume Problems 🎯

Workshop Complete!

Part 7: Comprehensive Assessment

Volumes of Revolution — Review

Part 7 of 7 — Comprehensive Assessment

Method	Formula
Disk	$V = \pi\int [R]^2\,dx$
Washer	$V = π$

R(x)

is the distance from the curve to the axis of rotation

Don't forget to square the radius AND multiply by

\pi

$R(x)$ = outer radius (farther from axis)
$r(x)$ = inner radius (closer to axis)

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

Outer: $R = x$ . Inner: $r = x^2$ .

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

Washer Method 🎯

Key Takeaways — Part 2

Washer method: $V = \pi\int [R^2 - r^2]\,dx$
Identify outer and inner radii carefully
The disk method is a special case where $r = 0$

$R(x) = |f(x) - k|$ (distance from outer curve to axis)
$r(x) = |g(x) - k|$ (distance from inner curve to axis)

Rotate the region between $y = x^2$ and $y = 1$ about $y = 2$ .

On $[-1, 1]$ : Outer radius: $R = 2 - x^2$ . Inner radius: $r = 2 - 1 = 1$ .

$V = \pi\int_{-1}^{1} [(2-x^2)^2 - 1^2]\,dx = 2\pi\int_0^1 (4-4x^2+x^4 - 1)\,dx$ $= 2\pi\int_0^1 (3-4x^2+x^4)\,dx = 2\pi\left[3x - \frac{4x^3}{3} + \frac{x^5}{5}\right]_0^1 = 2\pi\left(3 - \frac{4}{3} + \frac{1}{5}\right) = \frac{56\pi}{15}$

Rotation About Other Lines 🎯

Key Takeaways — Part 3

Radius = distance from curve to axis of rotation
When axis is below: $R = f(x) - k$
When axis is above: $R = k - f(x)$
Always think: what's farther (outer) vs closer (inner) to the axis

Common Cross-Section Formulas

If the base goes from $y = g(x)$ to $y = f(x)$ , the side length is $s = f(x) - g(x)$ .

Cross-section	Area Formula
Square	$A = s^2$
Semicircle	$A = \frac{\pi s^2}{8}$
Equilateral triangle	$A = \frac{\sqrt{3}}{4}s^2$
Isosceles right triangle (leg = side)	$A = \frac{s^2}{2}$

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

$s = \sqrt{x} - 0 = \sqrt{x}$ . $A = (\sqrt{x})^2 = x$ .

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

Cross-Section Volumes 🎯

Key Takeaways — Part 4

Cross-section problems: $V = \int A(x)\,dx$
Find the side length from the base region, then apply the area formula
This is a common AP free-response topic!

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

$x = \sqrt{y}$ , so $R = \sqrt{y}$ .

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

y-Axis Rotation 🎯

Key Takeaways — Part 5

For $y$ -axis rotation, express $x$ as a function of $y$
Use $\int dy$ with the same disk/washer formulas

V = π \int [R^{2} - r^{2}] d x

Final Assessment 🎯

Volumes of Revolution — Complete! ✅

You have mastered:

✅ Disk method (single curve rotation)
✅ Washer method (two curves, hole in middle)
✅ Rotation about non-standard axes
✅ Cross-sectional volumes with known shapes

2π[3x−34x3+5x5]01=

Volumes of Revolution

Volumes of Revolution - Complete Interactive Lesson

Part 1: Disk Method

Volumes of Revolution

Rotating Around the x-axis

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

Rotating About y=ky = ky=k or x=kx = kx=k

Part 4: Cross-Sectional Volumes

Volumes of Revolution

Known Cross-Sections

Part 5: Disk/Washer in y

Volumes of Revolution

Rotating About the yyy-axis Using dydydy

Part 6: AP-Style Workshop

Volumes of Revolution

Workshop Complete!

Part 7: Comprehensive Assessment

Volumes of Revolution — Review

Worked Example

Key Takeaways — Part 2

Worked Example

Key Takeaways — Part 3

Common Cross-Section Formulas

Worked Example

Key Takeaways — Part 4

Worked Example

Key Takeaways — Part 5

Volumes of Revolution — Complete! ✅

Rotating About $y = k$ or $x = k$

Rotating About the $y$ -axis Using $dy$