# Comparing Methods: Shell Method vs. Disk/Washer Method in Calculus

The **Shell method** and the **disk/washer method** are two important techniques in integral calculus used to find the volume of solids. As an expert in calculus and mathematics, I will **showcase my expertise ** by providing an in-depth comparison of these two methods – their concepts, procedures, applications, and relative advantages.

## Overview of the Two Methods

Before diving into the nitty-gritty details, let’s first outline what each method is at a high level:

### Shell Method

- Used to calculate the volume of a solid by using cylinders rather than washers or disks
- Involves integrating the height of the solid over its base area
- Often useful for solids with known cross-sections
- Relies on the geometric formula:
`Volume = Height x Base Area`

### Disk/Washer Method

- Calculates volume by integrating disks/washers perpendicular to an axis
- Disk method integrates disks while washer method integrates washers
- Useful when one of the bounding functions is rotational (e.g. y = f(x))
- Relies on the geometric formula:
`Volume = π x (Outer Radius)2 - π x (Inner Radius) 2`

Now let’s explore each of these methods more in-depth, including step-by-step procedures and examples.

## The Shell Method

The shell method calculator aims to calculate the volume of a solid by using cylinders rather than washers or disks. It involves integrating the height of the solid over its base area.

Here are the key steps to apply the shell method:

- Identify the solid and the axis of rotation. The axis of rotation will be parallel to the height of the cylinders used.
- Determine the bounds of integration. This involves identifying the endpoints where the height of the cylinders goes to 0.
- Identify a formula for the
**height**of the cylinders as a function of the axis of rotation. The height is usually the distance between the two functions bounding the solid. - Identify a formula for the
**base area**of the cylinders, which is based on the axis of rotation. - Integrate the height multiplied by the base area between the bounds. This gives the total volume.

The general formula is:

```
V = ∫ height(x) × base(x) dx
```

Where:

- V is the total volume
- height(x) is the height of the cylinders
- base(x) is the base area of the cylinders

Let’s look at an example solid and how to apply the shell method.

*Shell method example solid*

Here, the solid is bounded between y = x and y = x2 from x = 0 to x = 2.

- Axis of rotation: x-axis
- Bounds of integration: 0 to 2
- Height: Difference between functions = x2 – x = x(x – 1)
- Base: Circles with radius x
- Base area: πx2

Plugging this into the formula:

```
V = ∫ from 0 to 2 x(x - 1)πx2 dx
V = π ∫ from 0 to 2 x3(x - 1) dx
V = π [x4/4 - x3/3] from 0 to 2
V = π (16/4 - 8/3) = 16π/3
```

So the volume using shell method is 16π/3 cubic units.

Some key **benefits** of the shell method:

- Often simpler than disk or washer method for solids with known cross-sections
- Avoid dealing with multiple integrals like in disk/washer method
- Intuitive by visualizing stacking cylinders

However, it can only be used when it’s easy to represent the base area formula. The disk/washer method is more flexible since it can handle more complex solids.

## Disk and Washer Method

The **disk method** and **washer method** are very similar techniques used to calculate the volume of a solid when one of its bounding functions is rotational (e.g. y = f(x)).

- The disk method integrates disks perpendicular to the axis of rotation
- The washer method integrates washers perpendicular to the axis

While the formulas look different, the underlying concepts are analogous. The key differences come down to the setup of the integral.

### Key Steps

Here are the general steps applied in both methods:

- Identify the axis of rotation and the bounds of integration
- Determine the formulas for the outer and inner radii of the disks/washers
- Write an integral summing the volumes of the disks/washers
- Integrate to find the total volume

The general formulas are:

```
Disk method:
V = ∫ π (outer radius)2 dx
Washer method:
V = ∫ π (outer radius)2 - π (inner radius)2 dx
```

Where outer radius and inner radius depend on the bounds.

Let’s go through an example to compare.

*Diagram for disk vs washer example*

Find the volume of the solid bounded between the parabolas y = x2 and y = 4 – x2 from x = 0 to x = 2 revolved about the x-axis.

#### Disk Method

- Axis of rotation: x-axis
- Bounds of integration: 0 to 2
- Outer radius: x2
- No inner radius

*Formula*:

```
V = ∫ from 0 to 2 π (x2)2 dx
V = ∫ from 0 to 2 πx4 dx
V = π[x5/5] from 0 to 2
V = π(32/5) = 16π/5
```

#### Washer Method

- Axis of rotation: x-axis
- Bounds: 0 to 2
- Outer radius: 4 – x2
- Inner radius: x2

*Formula:*

```
V = ∫ from 0 to 2 π[(4 - x2)2 - (x2)2] dx
V = ∫ from 0 to 2 π(16 - 8x2 + x4) dx
V = π[16x - 4x3 + x5/5] from 0 to 2
V = π(32 - 16 + 16/5) = 16π/5
```

We get the same volume with both methods! The washer method accounts for the empty inner radius while disks simply integrate the outer radius.

Some **benefits** of this technique:

- Extremely useful for solids of revolution bounded by functions of x
- Avoid guesswork needed for other volume methods
- Can compare disk and washer approaches

The downside is dealing with multiple integrals, making it tougher than the shell method in some cases.

## Summary and Comparison Table

Here’s a table summarizing the key points on using the shell method vs. the disk/washer method:

**Shell Method** | **Disk/Washer Method**

|—|—|—|

**What is it**

- Uses cylinders perpendicular to rotational axis instead of washers/disks

- Uses disks or washers perpendicular to axis of rotation

**When to use**

- Solids with known cross sections
- Height and base area are simple to define

- One bounding function is rotational (y = f(x))
- Can’t easily apply shell method

**Typical Problems**

- Between functions of x
- Extruded shapes and cross-sections

- Functions rotated about x or y axes
- Solids of revolution problems

**Procedure** 1. Identify axis of rotation

2. Set bounds

3. Define height formula

4. Define base area formula

5. Integrate height*base | 1. Identify axis

2. Set bounds

3. Define radii

4. Integrate disks/washers

**Pros**

- Often simpler for certain solids
- Avoids multiple integrals
- Intuitive cylinders

- Extremely versatile
- Works for solids of revolution
- Can compare disks and washers

**Cons**

- Limited to known cross-sections
- Can’t handle as many solids

- More complex setup
- Multiple integral

So in summary, the shell method is best for solids with nice known cross sections, while disk and washer methods are more widely applicable, especially to solids of revolution – but involve more complex integrals.

Understanding when and how to apply each method takes practice across a diverse range of problems. As you tackle more volume problems, pay attention to whether cylinders or washers/disks align better with the bounding surfaces, and apply the appropriate technique.

## Applications and Examples

Now let’s look at some examples of applying these techniques:

### Finding the Volume of a Trough

*The cross-section trough bounded by y = 8 – x2 and y = 0 from x = 0 to x = 2*

Because we have a known cross section formula between two functions of x, the shell method is ideal here.

- Axis of rotation: x-axis
- Bounds: 0 to 2
- Height: 8 – x2
- Base: y = 0, so base radius is x
- Base area: πx2

*Integral:*

```
V = ∫ from 0 to 2 (8 - x2)πx2 dx
V = π∫ from 0 to 2 (8x2 - x4) dx
V = π[8x3/3 - x5/5] from 0 to 2
V = (32π/3) - (16π/5)
= 16π/3
```

### Volume of Solid of Revolution

*The region between y = 8 – x2 and y = 0, revolved about the y-axis from x = 0 to x = 2*

Since one bound involves revolving a function about the y-axis, washers would align perfectly to this solid.

- Axis of rotation: y-axis
- Bounds: 0 to 2
- Outer radius: 8 – x2
- Inner radius: 0

*Integral:*

```
V = ∫ from 0 to 2 π[(8 - x2)2 - 02] dx
V = ∫ from 0 to 2 π(8 - x2)2 dx
V = π∫ from 0 to 2 (64 - 16x2 + x4) dx
V = π[64x - 8x3 + x5/5] from 0 to 2
= (128π - 32π + 16π/5)
= 112π/5
```

### Volumes of Other Solids

These methods can be used to derive volumes of many other geometrical solids:

- Cones
- Pyramids
- Spheres
- Cylinders

The method choice depends on the alignment of the surfaces to washers/disks vs perpendicular cylinders.

For example, to calculate the volume of a cone with height *h* and radius *r*:

- Cylinders align perfectly along the height of the cone
- Use shell method with base area πr2
- Integrate height
*base from 0 to*h*

This avoids the more complex washer or disk integrals.

Understanding these subtle choices comes with practice across diverse problems, building intuition on the best techniques.

## Technique Comparison in Practice

While conceptually distinct, there is often overlap in how these techniques can be applied:

- Certain solids allow both methods
- Different setups may be easier with one vs the other

For example, in the solid of revolution problem earlier – if we know the cross-sectional area, we could also integrate it using the shell method along the y axis.

However, the washer method is more direct by aligning washers with the surfaces.

When multiple approaches are possible:

**Try both methods!**- See which integral is simpler
- Compare if the answers match

This helps develop flexibility in volume problems.

Here are some guidelines I recommend based on practice:

- If nice clear cross-sections available -> start with shell method
- If rotating a function -> try washers/disks first
- If stuck on one method -> try the other approach
- When possible, verify if both give the same volume!

Building intuition for when to use cylindrical shells vs washers/disks comes from problem-solving experience. So practicing across diverse problems is key.

## Wrapping Up

Calculating volumes in integrals often requires visualizing solids as stacks of simpler geometric pieces – cylinders, washers or disks. The shell method and disk/washer method provide two powerful techniques:

**Key points:**

- Shell method uses cylinders perpendicular to an axis
- Disk and washer methods use disks/washers perpendicular to an axis
- Shell method integrates height x base area
- Disk/washer integrates π x (outer – inner) radii2

**When to use:**

- Shell method -> Known cross-sections
- Disk/washer -> One function is rotational

Mastering volume calculation involves understanding when to align washers vs. cylinders to a particular solid based on its bounds.

Through numerous examples across various solids, these methods provide flexible tools to derive volumes in calculus and beyond.

I hope you found this overview useful! Let me know if you have any other questions.