The Disk Method Vs Washer Method: Which One To Choose?

Calculus is not a favorite subject for everyone. But, it is necessary to solve real-life problems related to volumes and areas. In technical terms, Disk Method and Washer Method are two techniques used to calculate integrals by revolving solid bodies around an axis. These two methods are the bread and butter of calculus when it comes to finding volumes of 2D shapes in 3D space. If you are looking for a clear comparison of the Disk Method vs Washer Method, then look no further than this article.

What is the Disk Method?

The Disk Method is a calculation method in calculus that is used to find the volume of a solid of revolution by adding up the volumes of several tiny disks. The disk method uses disk-shaped infinitesimal elements to calculate the volume. Consider a solid of revolution like a wine bottle. You can visualize that rotating the bottle around the central axis will create a structure that approximates a disc. If the wine bottle was sliced into very thin discs and the disks’ volumes added up, the result would be an approximation of the original solid of revolution’s volume.

When using the disk method, two cases can be considered. The first case is when the axis of revolution passes through the figure’s center of mass. The second case is when the axis of revolution does not pass through the figure’s center of mass.

What Is The Washer Method?

The washer method is similar to the disk method but results in a hollow shape with a hole in the center. Like the disk method, this calculation method uses circular plates stacked up along an axis; however, instead of using solid discs, the washer method uses cylindrical shells with interior and exterior radii. In simple terms, the washer method is used when the shape you are revolving around the axis has a hole (like a donut) or is not centered around the axis of revolution (like a bowl or flower vase).

If you rotate a flat shape like a rectangle around an axis, you end up with a cylinder. If you now remove a smaller cylinder from the center, you created a shape like a washer or donut.

Disk Method vs Washer Method

Both disk and washer methods are used to calculate the volume of a solid of revolution. However, they differ in the way they do the calculation method. The disk method involves cutting the solid of revolution perpendicular to its axis or revolving an area revolved around the axis to generate a 3D object made up of stacking disks. In contrast, the washer method involves cutting the solid of revolution perpendicular to its axis or revolving an area revolved around the axis to generate a 3D object made up of cylindrical shells.

Both methods have their uses, and it’s important to know which one to use based on the shape’s geometry. When the solid has no hole, the disk method is used; otherwise, the washer method is the way to go. In some cases, you might need to use both methods together. They differ in how they calculate the volume of a shape that is not uniform around the axis of rotation.

The disk method is mostly used to find the volume of a 3D shape by summing up the disk areas or volumes of the cross-sections perpendicular to the axis of revolution. It is very useful when dealing with solids of revolution, especially those with rotational symmetry.

The washer method is used to find the volume of a 3D shape that resembles a washer or donut. The volume is calculated by summing up the volumes of the hollow shells generated by rotating the function of the original 2D shape. This method is helpful when finding the volume of complex shapes that are not symmetric around the axis of revolution.

When to Use the Disk Method

The disk method is mostly used for finding the volumes of shapes that are symmetric around the axis of rotation. It is useful when dealing with cylinders and cones or when dealing with shapes generated by revolving simple curves around an axis. It is also handy when the solid of revolution has a known radius or height.

When to Use the Washer Method

The washer method is useful for solving problems related to shapes with holes in them. It is also very helpful for finding the volume of shapes that are not symmetric around the axis of revolution. This method is mostly used in dealing with shapes like bowls, vases or furniture legs, which have shapes that are not uniform.

In Conclusion

The Disk Method vs Washer Method debate boils down to the shape’s geometry that is being revolved around the axis of rotation. The disk method is used when the shape has no hole or when the shape has rotational symmetry, whereas the washer method is used when the shape has a hole or is not uniform around the axis of revolution. In general, it is advisable to use both methods judiciously rather than rely on one exclusively.