Unit 8.9 – Volume with Disc Method: Revolving Around the x- or y-Axis

AP® Calculus AB & BC | Solids of Revolution

Why This Matters: The disc method is THE fundamental technique for finding volumes of solids of revolution! When you rotate a region around an axis, you create a 3D solid. By slicing it perpendicular to the axis of rotation, each slice is a circular disc. The volume is the integral of all disc areas! This method is essential for AP® Calculus and appears on virtually every exam. Master this and you've unlocked one of calculus's most powerful applications!

🎯 The Disc Method Concept

THE BIG IDEA

When a region is rotated around an axis, it creates a solid. Slice the solid perpendicular to the axis of rotation, and each slice is a circular disc!

The Volume Formula:

Volume = Sum of all disc volumes = Integral of disc areas

\[ \text{Volume of one disc} = \pi r^2 \cdot \text{thickness} \]

\[ \text{Total Volume} = \int \pi r^2 \, dx \text{ or } \int \pi r^2 \, dy \]

↔️ Revolving Around the x-Axis

Disc Method: x-Axis

THE FORMULA:

When rotating region bounded by \(y = f(x)\), \(y = 0\), \(x = a\), and \(x = b\) around the x-axis:

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

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

Where:

\(R(x) = f(x)\) = radius of disc at position \(x\)
Radius = distance from x-axis to curve
\([a, b]\) = bounds of integration (x-values)
Slicing perpendicular to x-axis

📝 Key Insight: The radius at any point is simply the y-value of the function: \(R(x) = f(x)\). Don't forget to SQUARE it!

↕️ Revolving Around the y-Axis

Disc Method: y-Axis

THE FORMULA:

When rotating region bounded by \(x = g(y)\), \(x = 0\), \(y = c\), and \(y = d\) around the y-axis:

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

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

Where:

\(R(y) = g(y)\) = radius of disc at position \(y\)
Radius = distance from y-axis to curve
\([c, d]\) = bounds of integration (y-values)
Slicing perpendicular to y-axis
Must express function as \(x = g(y)\)

📋 Step-by-Step Process

Complete Method

The 6-Step Approach:

Sketch the region: Draw the area to be rotated
Identify axis of rotation: x-axis or y-axis?
Determine radius function: \(R(x)\) or \(R(y)\) = distance from axis to curve
Find limits of integration: Bounds along axis of rotation
Set up integral: \(V = \pi \int_a^b [R]^2 \, dx\) or \(dy\)
Evaluate: Expand, integrate, and simplify

📖 Comprehensive Worked Examples

Example 1: Around x-Axis (Basic)

Problem: Find the volume when the region bounded by \(y = \sqrt{x}\), \(y = 0\), \(x = 0\), and \(x = 4\) is rotated around the x-axis.

Solution:

Step 1: Identify setup

Rotating around x-axis

Bounds: \(x = 0\) to \(x = 4\)

Step 2: Find radius

Radius at position \(x\):

\[ R(x) = \sqrt{x} \]

Step 3: Set up integral

\[ V = \pi \int_0^4 (\sqrt{x})^2 \, dx = \pi \int_0^4 x \, dx \]

Step 4: Evaluate

\[ V = \pi \left[\frac{x^2}{2}\right]_0^4 = \pi \cdot \frac{16}{2} = 8\pi \]

ANSWER: \(V = 8\pi\) cubic units

Example 2: Around y-Axis

Problem: Rotate the region bounded by \(y = x^2\), \(y = 0\), and \(y = 4\) around the y-axis. Find the volume.

Step 1: Convert to x = g(y)

From \(y = x^2\):

\[ x = \sqrt{y} \]

(taking positive root since region on right)

Step 2: Setup

Rotating around y-axis

Bounds: \(y = 0\) to \(y = 4\)

Radius: \(R(y) = \sqrt{y}\)

Step 3: Integrate

\[ V = \pi \int_0^4 (\sqrt{y})^2 \, dy = \pi \int_0^4 y \, dy \]

\[ = \pi \left[\frac{y^2}{2}\right]_0^4 = 8\pi \]

Example 3: Polynomial Function

Problem: Find volume when region between \(y = x^2\) and \(y = 0\) from \(x = 0\) to \(x = 2\) is rotated around the x-axis.

Setup and solve:

\[ V = \pi \int_0^2 (x^2)^2 \, dx = \pi \int_0^2 x^4 \, dx \]

\[ = \pi \left[\frac{x^5}{5}\right]_0^2 = \pi \cdot \frac{32}{5} = \frac{32\pi}{5} \]

Example 4: Trigonometric Function

Problem: Rotate \(y = \sin x\) from \(x = 0\) to \(x = \pi\) around the x-axis.

Setup:

\[ V = \pi \int_0^\pi (\sin x)^2 \, dx = \pi \int_0^\pi \sin^2 x \, dx \]

Use identity: \(\sin^2 x = \frac{1 - \cos 2x}{2}\)

\[ V = \pi \int_0^\pi \frac{1 - \cos 2x}{2} \, dx = \frac{\pi}{2}\left[x - \frac{\sin 2x}{2}\right]_0^\pi \]

\[ = \frac{\pi}{2}[\pi - 0] = \frac{\pi^2}{2} \]

📊 Quick Comparison: x-Axis vs y-Axis

Disc Method Comparison
Feature	x-Axis Rotation	y-Axis Rotation
Formula	\(\pi \int_a^b [f(x)]^2\,dx\)	\(\pi \int_c^d [g(y)]^2\,dy\)
Radius	\(R(x) = f(x)\)	\(R(y) = g(y)\)
Variable	Integrate with \(dx\)	Integrate with \(dy\)
Bounds	x-values: \([a, b]\)	y-values: \([c, d]\)
Function Form	\(y = f(x)\)	\(x = g(y)\)

💡 Essential Tips & Strategies

✅ Success Strategies:

Sketch the region: Visualize what's being rotated
Identify the axis: x-axis or y-axis determines everything
Radius = distance from axis: To the outer curve
SQUARE the radius: Don't forget! Area = πr²
Keep π outside integral: Factor it out for cleaner work
For y-axis: Convert to x = g(y) first
Expand before integrating: Makes integration easier
Check units: Cubic units for volume

🔥 Common Setups:

Around x-axis: Use \(dx\), radius = y-coordinate
Around y-axis: Use \(dy\), radius = x-coordinate
√x squared = x: Simplify before integrating
Trig functions: Remember power-reduction formulas
Always include π: In final answer

❌ Common Mistakes to Avoid

Mistake 1: Forgetting to square the radius function
Mistake 2: Forgetting π in the formula
Mistake 3: Using wrong variable (dx when should use dy)
Mistake 4: Wrong limits of integration
Mistake 5: Not converting to x = g(y) for y-axis rotation
Mistake 6: Squaring incorrectly: (√x)² = x, not √(x²)
Mistake 7: Integration errors (especially with trig)
Mistake 8: Wrong axis of rotation (mixing up x and y)
Mistake 9: Arithmetic errors in simplification
Mistake 10: Not including units or leaving π inside

📝 Practice Problems

Find the volume:

Region bounded by \(y = x\), \(y = 0\), \(x = 3\) rotated around x-axis.
Region bounded by \(y = x^2\), \(y = 4\) rotated around x-axis.
Region bounded by \(x = y^2\), \(x = 4\) rotated around y-axis.
Region bounded by \(y = e^x\), \(y = 0\), \(x = 0\), \(x = 1\) around x-axis.

Answers:

\(9\pi\) cubic units
\(\frac{128\pi}{5}\) cubic units
\(\frac{32\pi}{5}\) cubic units
\(\frac{\pi(e^2-1)}{2}\) cubic units

✏️ AP® Exam Success Tips

What AP® Graders Look For:

State axis of rotation: Clearly identify x-axis or y-axis
Show radius function: Write \(R(x)\) or \(R(y)\)
Correct integral setup: \(V = \pi \int [R]^2\,dx\) or \(dy\)
Show squaring: Explicitly square the radius
Show integration work: Find antiderivative
Evaluate at bounds: Show substitution
Simplify answer: Factor out π, simplify fractions
Include units: "cubic units" or specific units

💯 Exam Strategy:

Read carefully: Which axis of rotation?
Sketch region if time permits
Identify radius function (distance from axis)
For y-axis: Convert to x = g(y)
Write setup: \(V = \pi \int_a^b [R]^2\,dx\) (or dy)
Square the radius explicitly
Expand before integrating
Evaluate and simplify
Include π in final answer

⚡ Quick Reference Guide

DISC METHOD ESSENTIALS

Around x-Axis:

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

Radius = \(f(x)\) (y-coordinate)

Around y-Axis:

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

Radius = \(g(y)\) (x-coordinate)

Remember:

Radius = distance from axis to curve
ALWAYS square the radius!
Include π in the formula
Match variable: dx with x-axis, dy with y-axis

Master the Disc Method! The fundamental formulas: around x-axis: \(V = \pi\int_a^b[f(x)]^2\,dx\) where radius \(R(x) = f(x)\) is the y-coordinate; around y-axis: \(V = \pi\int_c^d[g(y)]^2\,dy\) where radius \(R(y) = g(y)\) is the x-coordinate (must convert to \(x = g(y)\) form). The disc method: rotate region around axis → creates 3D solid → slice perpendicular to axis → each slice is circular disc with area \(\pi r^2\) → integrate disc areas. Critical: SQUARE the radius function! Common setup: identify axis of rotation, find radius = distance from axis to curve, set up \(\pi\int[R]^2\), expand and integrate. For y-axis problems, convert \(y = f(x)\) to \(x = g(y)\) first. Units: cubic units. This is a major AP® topic—appears every year! Practice both x-axis and y-axis rotations until automatic! 🎯✨