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 $x = 4$ is rotated about the $x$-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

1. Disk method: $V = \pi\int_a^b [R(x)]^2\,dx$
2. $R(x)$ is the distance from the curve to the axis of rotation
3. Don't forget to square the radius AND multiply by $\pi$

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):

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

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

Worked Example

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

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

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):

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

Worked Example

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

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

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

Common Cross-Section Formulas

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

Worked Example

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

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

Volumes of Revolution

Part 5 of 7 — Disk/Washer in $y$

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

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

Worked Example

$y = x^2$ 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

Volumes of Revolution

Part 6 of 7 — AP-Style Workshop

AP-Style Volume Problems 🎯

Workshop Complete!

Volumes of Revolution — Review

Part 7 of 7 — Comprehensive Assessment

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