Calculus is a mathematical field that deals with the study of change. It is among the most challenging and abstract fields in modern mathematics and is used in nearly every science, engineering, and business area.
Calculus helps us model situations where we have rates of change, such as velocity or acceleration. These are often called “differential equations.” Calculus also allows us to solve problems that involve limits: for example, finding the area under a curve or the volume of a solid.
You can use different methods to solve different problems. A few of these methods include the disk, washer, and shell methods.
The main difference between the disk, washer, and shell methods in calculus is that they all use different approaches to approximating a curve. The disk method uses a circular region around an approximation of the curve, while the washer uses a region shaped like a washer when viewed from above. The shell method uses a region shaped like a shell when viewed from above.
Let’s discuss all these methods in detail.
What Is Meant By The Disk Method?
The disc integration method, also known as integral calculus’ disc equation, calculates the volume of a solid per revolution when integrated along the axis parallel to its revolution.
The disk method involves dividing an object into many small disks or cylinders and then adding the volumes of these small disks together to determine the object’s volume.
A cylinder’s radius is given by a function f(x), and its height is determined by x. When the change in x reaches zero and the number of disks increases to infinity, you will have the actual volume of the object rather than an estimate.
The formula for calculating volume through the disk integration method is as follows:
 | = | distance between the function and the axis of rotation |
 | = | upper limit |
 | = | lower limit |
 | = | slides along x |
What Is Meant By The Washer Method?
The washer method is a way to solve a differential equation. It’s called the washer method because it uses a washer as an analogy for how it works.
A differential equation describes how an unknown function changes as time passes, even if it’s not continuous. It’s often used to model things like waves or other processes that change over time, but not necessarily in a smooth way.
To solve for y(t), you need to find y(t) for all possible values of t. However, this can be difficult and time-consuming because there are infinite solutions. The Washer Method helps you find solutions using approximations instead of exact values.
- It starts with an initial guess at what your solution might look like y(t) = f(t).
- Then you find the error between this guess and what happens: e(t).
- You then use this error term to update your guess: f'(t) = f*2 – 2f*e + c, where c is an arbitrary constant (it doesn’t matter what value you choose).
- Then repeat the process until the error becomes smaller than epsilon.
What Is Meant By the Shell Method?
In calculus, the shell method is a technique for finding the volume of a solid by approximating it with a series of concentric shells. It is often used to find the volume of an irregularly shaped solid that cannot be easily partitioned into simple shapes for which the volumes are known.
The shell method divides the shape into many thin slices and then sums up all their volumes. The slices can be considered shells, hence the “shell method.”
The shell method differs from other methods by choosing a point as the center of the shell instead of the midpoint of each subinterval as the center. This results in more accurate approximations than other methods but requires more work on the user’s end.
Know The Difference
Shell, washer, and disc methods are all ways to solve calculus problems involving integration.
The shell method involves finding the volume of an annulus, while the disc method involves finding the area under a function’s curve. A washer method is similar to a shell method, but it uses a different technique to find the volume of an annulus.
Shell Method
The shell method is used to approximate the volume of a solid in revolution with a specified cross-section by summing the volumes of an infinite number of thin shells cut from the solid. The shell method is only valid when the cross-section has a constant thickness, so it cannot be used to find the volume of an irregularly shaped object.
Washer Method
The washer method is similar to the shell method except that instead of cutting an infinite number of thin shells from the solid, you cut just one thick shell from it (that has constant thickness) and then partitioned it into smaller pieces with a constant width.
Disc Method
The disc method involves drawing a series of circles with varying radii and different angular positions around an axis passing through their centers; these circles intersect at points that must lie on each other’s perimeters—in other words, they overlap—to form sectors that represent parts of a circle’s circumference.
These sectors are then added up to get an approximation of how many times each radius will fit around your object’s perimeter before overlaps occur between them all again at their following intersections along those same axes.
Comparison of Shell Method, Washer Method, and Disc Method
The table gives you the difference between the three methods in summarized form.
| Shell Method | Washer Method | Disc Method |
| The shell method works by slicing the solid object into thin slices and adding their areas. | The washer method works by slicing the solid object into thin slices and adding up their volumes. | The disc method works by taking a circle with a radius equal to the distance between two points on opposite sides of an arc and adding up all the area within that arc. |
Here is a video clip explaining all three methods.
When Should You Use The Washer Method Or The Shell Method?
Several methods exist for calculating the surface area of a cylinder. The shell method is one of them, but it’s not always the most efficient or accurate way.
The washer method isn’t a method—it’s just another way of saying, “What’s left over when you do this other thing?” It doesn’t tell you anything about what happens inside the cylinder; only what’s extraneous matters.
So which should you use? It depends on what you’re trying to measure!
If you want to know how much paint would be needed for your walls, the shell method will give you better results than the washer method because it uses more data points. But if you’re trying to measure how much rubber your tires require, the washer method will work better because it uses fewer data points.
How Do You Know If It’s A Disk Or A Washer?
The difference between a washer and a disk lies in their degree of rotational symmetry. A disk has no axis of symmetry, so it can be rotated through any angle and appear the same. A washer, however, has an axis of symmetry—a line that aligns the two halves of the object.
In calculus, you can tell the difference between a disk and a washer using the following equation:
Disk: (diameter)2 – (radius)2 = area of the disk
Washer: (diameter)2 < (radius)2
Final Thoughts
- Calculus offers methods like disk, washer, and shell for volume calculations.
- Each method has a unique approach to approximating solid object volumes.
- The disk method divides the objects into disks and sums their volumes.
- The washer method slices the objects into washers and adds their volumes.
- The shell method sums volumes of thin shells cut from the solid.
- The choice depends on object shape and calculation efficiency.
- Understanding these differences is crucial for accurate problem-solving.
- Each method has pros and cons. They are suitable for different calculus scenarios.