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Disc Method
Revolving Around Axes

Learn to compute volumes of revolution by slicing solids into infinitely thin circular discs and integrating their areas along the x-axis or y-axis.

AP Calculus BC

What Is a Solid of Revolution?

When you take a region in the plane and spin it around an axis, the resulting 3D shape is called a solid of revolution. Think of a potter's wheel: the profile curve sweeps out a full 360° to create the pot.

x axis of rotation profile curve circular disc

The Key Idea: Thin Circular Discs

Each disc is a cylinder with radius equal to the function value and thickness \( dx \) (or \( dy \)). The volume of one thin disc is:

x \( r(x) \) \( dx \) circular face Area = \( \pi r^2 \)

Summing infinitely many such discs from \( x = a \) to \( x = b \) gives the total volume.

Why the Disc Method Matters

AP Exam Staple
Volumes of revolution appear on nearly every AP Calculus exam, both AB and BC.
Engineering Applications
Designing tanks, pipes, turbine blades, and any rotationally symmetric object.
Foundation for Washer/Shell
Mastering discs is the prerequisite for the washer and shell methods.

Disc Formula: Revolving Around the x-Axis

x y a b \( f(x) \) \( y = f(x) \)

The radius of each disc equals \( f(x) \). Squaring it gives the disc area \( \pi[f(x)]^2 \), and integrating sums all disc volumes from \( a \) to \( b \).

A transparent lathe reveals the hole inside a washer

A hollow wooden bowl on a stationary lathe accompanies a calibrated generating region and a separate annular cross-section.Rotating the region from y = 0 to y = x², for 0 ≤ x ≤ 2, about the x-axis gives disks with radius x². The volume is π∫₀²x⁴ dx = 32π/5.

The axis determines the radius

Rotating the region from y = 0 to y = x², for 0 ≤ x ≤ 2, about the x-axis gives disks with radius x². The volume is π∫₀²x⁴ dx = 32π/5.Rotating the region from y = 0 to y = x², for 0 ≤ x ≤ 2, about the x-axis gives disks with radius x². The volume is π∫₀²x⁴ dx = 32π/5.

Disc Formula: Revolving Around the y-Axis

x y c d \( g(y) \) \( x = g(y) \)

When revolving around the y-axis, rewrite the curve as \( x = g(y) \). The radius of each disc is \( g(y) \), and we integrate with respect to \( dy \).

x-Axis vs y-Axis Revolution

Revolve Around x-Axis Revolve Around y-Axis
Integrate w.r.t. \( dx \) \( dy \)
Radius is \( f(x) \) \( g(y) \)
Formula \( \pi\!\int_a^b [f(x)]^2\,dx \) \( \pi\!\int_c^d [g(y)]^2\,dy \)
Discs perpendicular to x-axis (vertical slices) y-axis (horizontal slices)
Must express curve as \( y = f(x) \) \( x = g(y) \)

How to Apply the Disc Method

1
Sketch the region and identify the axis of revolution.
2
Express the radius as a function of the integration variable: \( f(x) \) for x-axis, \( g(y) \) for y-axis.
3
Determine the limits of integration \( [a, b] \) or \( [c, d] \).
4
Set up the integral: \( V = \pi \int [\text{radius}]^2 \, d(\text{variable}) \).
5
Evaluate the integral and include the \( \pi \) factor.

Example 1: \( y = \sqrt{x} \) Around the x-Axis

Find the volume when the region under \( y = \sqrt{x} \) from \( x = 0 \) to \( x = 4 \) is revolved about the x-axis.

x y 0 4 2 \( y=\sqrt{x} \)

Example 2: \( y = x^2 \) Around the y-Axis

Find the volume when the region bounded by \( y = x^2 \), \( y = 0 \), and \( y = 9 \) is revolved about the y-axis.

Example 3: \( y = \sin x \) Around the x-Axis

Find the volume obtained by revolving the region under \( y = \sin x \) from \( x = 0 \) to \( x = \pi \) about the x-axis.

x y 0 \( \frac{\pi}{2} \) \( \pi \) 1 \( y = \sin x \)

When to Rewrite the Function

The axis of revolution determines whether you need \( y = f(x) \) or \( x = g(y) \). If the problem gives you one form but you need the other, solve for the opposite variable.

Given \( y = x^3 \)
Revolve around y-axis: rewrite as \( x = y^{1/3} \), then use \( g(y) = y^{1/3} \).
Given \( x = e^y \)
Revolve around x-axis: rewrite as \( y = \ln x \), then use \( f(x) = \ln x \).

Visualizing the Radius

The radius of each disc is always the perpendicular distance from the curve to the axis of revolution. Drawing this line segment on your sketch is the most important step.

x-axis revolution \( r = f(x) \) x-axis y-axis revolution \( r = g(y) \) y-axis

Why \( \pi r^2 \) Works: The Riemann Sum

The disc method is a direct application of the Riemann sum. We partition \( [a, b] \) into \( n \) subintervals of width \( \Delta x \). Each slice is approximately a cylinder:

Each term \( \pi [f(x_i^*)]^2 \Delta x \) is the volume of a thin disc: \( \pi r^2 \) for the circular face area times \( \Delta x \) for the thickness. The integral sums these infinitesimally thin discs exactly.

Common Mistakes to Avoid

✕ Forgetting to Square
Writing \( \pi \int f(x)\,dx \) instead of \( \pi \int [f(x)]^2\,dx \). The radius must be squared.
✕ Wrong Variable
Using \( dx \) when revolving around the y-axis. Match the variable of integration to the axis.
✕ Forgetting \( \pi \)
Computing the integral correctly but omitting the \( \pi \) factor out front.
✕ Wrong Limits
Using x-limits when integrating in \( dy \), or vice versa. Limits must correspond to the integration variable.
✕ Disc vs Washer
Using the disc formula when there's a gap between the region and the axis. That requires the washer method.

Summary: Disc Method at a Glance

Element Detail
What it computes Volume of a solid of revolution
Cross-section shape Full circle (disc)
x-axis formula \( V = \pi \int_a^b [f(x)]^2\,dx \)
y-axis formula \( V = \pi \int_c^d [g(y)]^2\,dy \)
When to use Region touches the axis of revolution (no gap)
When NOT to use Gap between region and axis: use washer method

Question 1 of 5

The region under \( y = x^2 \) from \( x = 0 \) to \( x = 3 \) is revolved about the x-axis. Which integral gives the volume?

Question 2 of 5

What is the volume when \( y = \sqrt{x} \) from \( x = 0 \) to \( x = 4 \) is revolved about the x-axis?

Question 3 of 5

The region bounded by \( y = x^3 \), \( y = 0 \), and \( x = 2 \) is revolved about the x-axis. Which integral gives the volume?

Question 4 of 5

When revolving around the y-axis, the variable of integration should be:

Question 5 of 5

Find the volume when \( y = e^x \) from \( x = 0 \) to \( x = 1 \) is revolved about the x-axis.