# Volumes of Revolution

Mathematics I · Integral Calculus · https://tryals.app/learn/mathematics-i/volumes-of-revolution

## Solids of Revolution & Disks

A **solid of revolution** is formed by spinning a plane region around an axis. We find its volume by slicing it into thin pieces and integrating.

The **disk method** slices **perpendicular** to the axis of rotation. Each slice forms a **disk** of radius $r(x)$ and thickness $dx$.

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

For a region with a hole, the **washer method** subtracts the inner hole using outer radius $R(x)$ and inner radius $r(x)$:

| Method | Formula | Slicing Orientation |
| --- | --- | --- |
| **Disk** | $\pi \int [r(x)]^2 dx$ | **Perpendicular** |
| **Washer** | $\pi \int (R^2 - r^2) dx$ | **Perpendicular** |

> **Common pitfall:** Forgetting to **square the radius** in the disk method. A disk's area is $\pi r^2$, so the radius must be squared.

## The Shell Method

The **shell method** slices **parallel** to the axis of rotation, cutting the solid into cylindrical **shells**. Unrolling a shell of radius $x$ and height $f(x)$ gives its volume.

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

Disks and shells are two ways to slice the **same** solid. Choose whichever matches the geometry to keep your integral simple.

| Feature | Disk / Washer | Shell Method |
| --- | --- | --- |
| **Slicing** | **Perpendicular** | **Parallel** |
| **Radius** | $r(x)$ from axis | $x$ from axis |

> **Common pitfall:** Mixing up disk and shell setups, or dropping the $2\pi x$ factor in shells. Always sketch a typical slice first.

## Practice questions

4 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. The disk method slices perpendicular to the axis of rotation; the shell method slices parallel to it. Why does having two methods help, when both compute the same volume?

A. Because the slicing direction decides which variable you integrate in.
B. Because the shell method is more accurate for solids with holes, where disks give an approximation.
C. Because the two methods apply to different solids and never both apply to the same one.
D. Because the disk method computes volume and the shell method computes surface area.

**Answer:** A. Because the slicing direction decides which variable you integrate in.

**Why:** Both are exact; the difference is entirely practical. Rotate y = x³ about the y-axis and shells give radius x and height x³, integrated in x; disks force you to invert to x = y^(1/3) and integrate in y with extra care at the boundary. Sometimes only one avoids splitting the solid, the same judgement as choosing u in parts, made for the same reason.

Page: https://tryals.app/practice/mathematics-i/volumes-of-revolution/the-disk-method-slices-perpendicular-to-the-axis-of-rotation-the

### 2. Rotating the region under y = 1/x from 1 to infinity gives **Gabriel's horn**: finite volume, infinite surface area. You could fill it with paint but never paint it. Where does the paradox dissolve?

A. In the analogy: filling uses volume, painting assumes constant thickness
B. In the geometry, an infinitely long solid cannot have finite volume
C. In the calculation, one of the two integrals is evaluated incorrectly
D. Nowhere, it is a genuine contradiction in the theory of integration

**Answer:** A. In the analogy: filling uses volume, painting assumes constant thickness

**Why:** Both integrals are right and they measure different things: volume converges like 1/x² and surface like 1/x, which diverges. The paint story smuggles in a constant thickness, and far along the horn that coat is thicker than the horn is wide, so the paradox is in the metaphor, not the mathematics.

Page: https://tryals.app/practice/mathematics-i/volumes-of-revolution/rotating-the-region-under-y-1-x-from-1-to-infinity-gives-gabriels

### 3. Select every statement about volumes of revolution that is true.

A. Disks are stacked along the axis; shells are wrapped around it.
B. Shells and disks give different volumes for the same solid.
C. The disk method integrates pi times radius squared.
D. The volume element for a shell is 2 pi times radius times height times thickness.
E. A washer is a disk with a concentric hole removed.

**Answer:** A. Disks are stacked along the axis; shells are wrapped around it.; D. The volume element for a shell is 2 pi times radius times height times thickness.; E. A washer is a disk with a concentric hole removed.

**Why:** Shells slice _parallel_ to the axis, not perpendicular, and a washer uses $R^{2} - r^{2}$ rather than $(R-r)^{2}$. The first three are exact; the fact that both methods describe the same solid is what lets you pick whichever integral is easier.

Page: https://tryals.app/practice/mathematics-i/volumes-of-revolution/select-every-statement-about-volumes-of-revolution-that-is-true

### 4. Order the steps for computing a volume of revolution.

**Answer:**

1. Sketch the region and identify the axis of rotation
2. Choose disks or shells according to how the slices meet the axis
3. Write the radius (and height, for a shell) as a function of the variable
4. Assemble the volume element for one slice
5. Integrate the volume element between the limits of the region

**Why:** The pattern is the same as Riemann sums: slice, express one slice's volume, integrate. Every setup error in this lesson happens in the middle steps, misidentifying the radius or attaching the wrong volume element to the slice.

Page: https://tryals.app/practice/mathematics-i/volumes-of-revolution/order-the-steps-for-computing-a-volume-of-revolution