# Disc method around the x- or y-axis

> AP Calculus BC · Unit 8: Applications of Integration
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u8-disc-method-around-the-x/

This guide covers the disc (disk) method for calculating volumes of solids of revolution around the x-axis and y-axis for AP Calculus BC, including formula application, worked examples, and common exam pitfalls to avoid.

**Prerequisites:** Evaluating definite integrals analytically and numerically; Finding intersection points of two curves; Rewriting functions as $x$ in terms of $y$

## Learning objectives

- Explain the core intuition behind the disc method for volumes of revolution
- Apply the disc method for rotation around the x-axis and y-axis
- Find correct integration bounds from intersecting boundary curves
- Identify and avoid common exam errors in disc method problems

## Core Concept of the Disc Method

The disc method (alternately spelled disk method, both are acceptable on the AP exam) is an integration technique to calculate the volume of a 3D solid formed by rotating a 2D region around a coordinate axis (the x-axis or y-axis).

The method gets its name from its core approach: we slice the solid perpendicular to the axis of rotation, so each slice is a thin circular disc. We sum the volumes of all thin discs via integration to get the total volume of the solid.

> **info**
>
> Unlike the washer method, which handles solids with a central hole, the disc method only applies to solid solids of revolution with no internal cavity between the rotated region and the axis of rotation. This topic is part of Unit 8, which makes up 10–15% of your total AP exam score.

## Disc Method for Rotation Around the x-Axis

When rotating a region around the x-axis (a horizontal axis), we slice perpendicular to the x-axis, so each slice has a thickness of $dx$. The radius of any disc at position $x$ is the vertical distance from the x-axis (the axis of rotation, $y=0$) up to the boundary curve $y = f(x)$.

$$V = \pi \int_{a}^{b} \left[f(x)\right]^2 dx$$

For a region bounded by $y = f(x)$, the x-axis, $x=a$, and $x=b$, this formula gives the total volume of revolution. No reworking of the original function is required here, since we integrate directly with respect to $x$.

**Worked example:** Find the volume of the solid formed when the region bounded by $y = \sqrt{x}$, the x-axis, and $x=4$ is rotated around the x-axis.

1. Identify bounds of integration: The region starts at $x=0$, where $y = \sqrt{x}$ meets the x-axis, and ends at the given boundary $x=4$, so $a=0$, $b=4$.
2. Identify the radius: At any $x$, the distance from the x-axis to the curve is $r(x) = f(x) = \sqrt{x}$.
3. Substitute into the disc method formula:
4. $$V = \pi \int_{0}^{4} \left(\sqrt{x}\right)^2 dx = \pi \int_{0}^{4} x dx$$
5. Evaluate the definite integral:
6. $$\pi \left[\frac{1}{2}x^2\right]_0^4 = \pi \left(\frac{1}{2}(16) - 0\right) = 8\pi$$
7. The total volume of the solid is $8\pi$ cubic units.

> **Exam tip:** On AP FRQ, always write out the full integral setup before evaluating. You will earn most points for the correct setup, even if you make a small arithmetic error in the final calculation.

## Disc Method for Rotation Around the y-Axis

When rotating a region around the y-axis (a vertical axis), the disc method requires slicing perpendicular to the y-axis, so each slice has thickness $dy$, and we integrate with respect to $y$. This means we must rewrite the boundary function as $x = g(y)$, a function of $y$, because the radius of the disc at position $y$ is the horizontal distance from the y-axis (the axis of rotation, $x=0$) out to the curve.

$$V = \pi \int_{c}^{d} \left[g(y)\right]^2 dy$$

> **tip**
>
> A common point of confusion: the shell method for rotation around the y-axis integrates with respect to $x$, but the disc method always integrates with respect to the same variable as the axis of rotation. This means disc method around y-axis *always* requires rewriting the function in terms of $y$.

**Worked example:** Find the volume of the solid formed when the region bounded by $y = x^3$, the y-axis, and $y=8$ is rotated around the y-axis.

1. Rewrite the function as $x$ in terms of $y$: $y = x^3 \implies x = y^{1/3} = g(y)$.
2. Identify bounds of integration: The region runs from $y=0$, where the curve meets the y-axis, to the given boundary $y=8$, so $c=0$, $d=8$.
3. Radius at any $y$ is $r(y) = g(y) = y^{1/3}$, the distance from the y-axis to the curve.
4. Substitute into the formula:
5. $$V = \pi \int_{0}^{8} \left(y^{1/3}\right)^2 dy = \pi \int_{0}^{8} y^{2/3} dy$$
6. Evaluate the integral:
7. $$\pi \left[\frac{y^{5/3}}{5/3}\right]_0^8 = \pi \cdot \frac{3}{5} (8^{5/3}) = \frac{3\pi}{5}(32) = \frac{96\pi}{5}$$
8. The total volume of the solid is $\frac{96\pi}{5}$ (or $19.2\pi$) cubic units.

> **Exam tip:** If your function cannot be easily rewritten as $x$ in terms of $y$, the disc method is not the most efficient approach for rotation around the y-axis, and you should use the shell method instead. AP problems will never force you to use disc when it is impractical.

## Finding Bounds from Intersecting Curves

Most AP exam problems do not give you explicit bounds for integration. Instead, you are given a region bounded by two or more curves, and you must first find their intersection points to get the limits of integration. For the disc method, the region must be bounded on one side by the axis of rotation and on the other side by the curve, so intersection points give your bounds.

> **warning**
>
> Always confirm the region is between the axis of rotation and the curve: if there is a gap between the region and the axis, you have a hole, and you need the washer method instead of disc.

**Worked example:** Find the volume of the solid formed when the region bounded by $y = 4 - x^2$ and the x-axis is rotated around the x-axis.

1. Find bounds by solving for where the curve intersects the x-axis ($y=0$): $0 = 4 - x^2 \implies x = -2$ and $x = 2$, so $a=-2$, $b=2$.
2. Confirm the region is between the x-axis (axis of rotation) and the curve: For all $x$ between $-2$ and $2$, $y = 4 - x^2$ is non-negative, so the entire region sits between $y=0$ (axis) and the curve, so disc method applies.
3. Radius at any $x$ is $r(x) = 4 - x^2$, so the volume integral is:
4. $$V = \pi \int_{-2}^{2} (4 - x^2)^2 dx$$
5. Simplify and evaluate: $(4-x^2)^2 = 16 - 8x^2 + x^4$
6. $$V = \pi \left[16x - \frac{8}{3}x^3 + \frac{1}{5}x^5\right]_{-2}^{2} = \frac{512\pi}{15}$$
7. The total volume of the solid is $\frac{512\pi}{15}$ cubic units.

> **Exam tip:** If the region is symmetric across the y-axis (like this example), you can integrate from $0$ to $2$ and double the result to cut your calculation time in half.

## Common pitfalls

- **Wrong:** Rotating around the y-axis with the disc method, integrate with respect to x using the original $y=f(x)$ function, leaving the integral as $V = \pi \int_{a}^{b} [f(x)]^2 dx$.
  - Why it fails: Confuses disc method around y-axis with shell method around y-axis, which does integrate with respect to x.
  - Correct: Always rewrite your function as x in terms of y, set up the integral with respect to y, and confirm all terms are in y before integrating.
- **Wrong:** Forgets to square the radius when setting up the integral, writing $V = \pi \int_{a}^{b} f(x) dx$ instead of $V = \pi \int_{a}^{b} [f(x)]^2 dx$.
  - Why it fails: Confuses the area of a circle ($\pi r^2$) with the circumference ($2\pi r$) from the shell method.
  - Correct: Always add the step "Area of disc = πr²" to your FRQ setup to remind yourself to square the radius.
- **Wrong:** Uses the disc method when the region being rotated does not touch the axis of rotation, leading to an undercalculated volume.
  - Why it fails: Forgets that disc method is for solid cross-sections with no hole; a gap between the region and axis creates a hole that needs the washer method.
  - Correct: Before setting up the integral, check if there is a gap between the region and the axis between your bounds; if there is a gap, use washer method instead of disc.
- **Wrong:** Uses x-bounds for a disc method integral around the y-axis.
  - Why it fails: Mixes up which variable matches which axis of rotation for disc method.
  - Correct: Follow the rule: for disc method, integrate with respect to the same variable as the axis of rotation: x-axis → integrate dx, y-axis → integrate dy.
- **Wrong:** Squares a binomial incorrectly, writing $(x + 2)^2 = x^2 + 4$ instead of $x^2 + 4x + 4$.
  - Why it fails: Forgets the middle term when expanding squared binomials, a common algebraic error.
  - Correct: Always expand squared binomials step by step, or use your calculator to evaluate the integral of the unexpanded squared function to avoid this error.
- **Wrong:** Squares the formula's π when the radius already contains a π, writing $V = \pi^2 \int f(x)^2 dx$ when $r(x) = \pi f(x)$.
  - Why it fails: Confuses the constant π from the area formula with constants inside the radius.
  - Correct: Separate the formula's π from any internal constants: if $r(x) = \pi f(x)$, then $V = \pi (\pi f(x))^2 dx = \pi^3 \int f(x)^2 dx$.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| Volume: rotation around x-axis (disc) | $V = \pi \int_a^b [f(x)]^2 dx$ | Applies when region is bounded by $y=f(x)$, x-axis, $x=a$, $x=b$; integrate with respect to x. |
| Volume: rotation around y-axis (disc) | $V = \pi \int_c^d [g(y)]^2 dy$ | Applies when region is bounded by $x=g(y)$, y-axis, $y=c$, $y=d$; rewrite function as x in terms of y, integrate with respect to y. |
| Radius for x-axis rotation | $r(x) = f(x)$ | Distance from x-axis ($y=0$) to the curve $y=f(x)$. |
| Radius for y-axis rotation | $r(y) = g(y)$ | Distance from y-axis ($x=0$) to the curve $x=g(y)$. |
| Finding bounds from intersections | Set $f(x) = 0$ for x-axis intersection, $g(y) = 0$ for y-axis intersection | Bounds are the coordinates of the intersection points of the boundary curves. |
| Symmetry shortcut | $V = 2\pi \int_0^b [f(x)]^2 dx$ for symmetric region $-b$ to $b$ | Cuts calculation time in half on the exam. |
| Mass for uniform density | $M = \rho V$ | Applies to real-world problems with constant density, where $\rho$ is density per unit volume. |

## What's next

Mastering the disc method is an essential prerequisite for the washer method, which extends the disc method to solids of revolution with a central hole, and the cylindrical shells method, another common approach to calculating volumes of revolution around the x- and y-axis. Without correctly setting up disc method integrals and understanding the core slicing intuition, you will struggle to distinguish between disc, washer, and shell methods, a commonly tested skill on both AP Calculus MCQ and FRQ sections. This topic is part of the larger unit on applications of integration, which also includes finding arc length, surface area of revolution, and work done by a variable force, all of which rely on the same slicing-and-summing-via-integration framework you learned here.

- [Disc method around other axes](https://www.owlsprep.com/study/ap-calculus-bc-u8-disc-method-around-other-axes/)
- [Washer method around the x- or y-axis](https://www.owlsprep.com/study/ap-calculus-bc-u8-washer-method-around-the-x/)
- [Washer method around other axes](https://www.owlsprep.com/study/ap-calculus-bc-u8-washer-method-around-other-axes/)

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