Volumes of Revolution

Volumes of Revolution

Part 1 of 7 — The Disk Method

Topic Overview

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

Worked Example

Rotate $y = \sqrt{x}$ about the $x$-axis from $x=0$ to $x=4$.

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

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

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: $V = \pi\int_a^b R^2\,dx$
• $R$ = distance from curve to axis of rotation
• Don’t forget to square $R$ and include $\pi$
• Works when the region touches the axis (no hole)

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):

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

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

Worked Example

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).

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

Disk vs Washer

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 $r = 0$

Volumes of Revolution

Part 3 of 7 — Rotation About Other Lines

Adjusting Radii for Non-Standard Axes

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

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

Quick Reference

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?

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:

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

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 $s = f(x) - g(x)$:

Worked Example

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

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 $dy$:

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

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

When to Use $dy$

Worked Example

$y = x^2$ 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. 🔍

Compute. ✍️

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 $dy$

Volumes of Revolution

Part 6 of 7 — AP-Style Workshop

Typical AP FRQ Structure

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

Worked AP Problem

$R$ 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)

Volumes of Revolution

Part 7 of 7 — Comprehensive Assessment

Complete Formula Reference

Top AP Mistakes

Quiz — Methods 🎯

Quiz — Cross-Sections & Setup 🎯

Final classification. 🔍

Final Challenge. ✍️

Volumes of Revolution — Complete!

You’ve mastered:

You’re ready for AP-level volume problems!