Disk Vs Shell Method: An In-Depth Comparison

When it comes to solving complicated math problems or calculating volumes of irregular shapes, two methods often come into play: the Disk method and the Shell method. Both methods are based on the concept of integrals, and allow us to calculate the volume of complex shapes. In this article, we will explore the Disk Vs Shell Method, highlighting their similarities and differences, and also explaining when each method should be used.

Disk Method: Overview and Applications

The Disk method, also known as the Washer method or the Circular method, is used to calculate the volume of a solid created by revolving a two-dimensional shape around an axis. To use this method, we need to split the shape into small, disk-shaped elements, find the area of each disk, and add up the volume of all the disks to get the total volume of the solid.

The Disk method is most commonly used to calculate the volumes of solids that are formed by revolving a circle, a region bounded by a function, or an area between the curves around an axis. For example, consider a solid that is formed by revolving the area bounded by the curves y=2x^2, y=0, and x=1 around the x-axis. Using the Disk method, we can split the shape into small disks of radius x, find the area of each disk (which is πx^2), and integrate the resulting expression from 0 to 1 to get the total volume of the solid.

Shell Method: Overview and Applications

The Shell method, also known as the Cylinder method or the Horizontal method, is used to calculate the volume of a solid created by revolving a two-dimensional shape around an axis. However, instead of splitting the shape into disks, we split it into vertical, cylindrical shells. To use this method, we need to find the radius and height of each shell, and add up the volume of all the shells to get the total volume of the solid.

The Shell method is most commonly used to calculate the volumes of solids that are formed by revolving a region bounded by two curves around an axis. For example, consider a solid that is formed by revolving the area bounded by y=x^3, y=0, x=1, and x=2 around the y-axis. Using the Shell method, we can split the shape into small cylindrical shells of radius x and height (2-x^3), integrate the resulting expression from 0 to 1 to get the volume of the first half of the solid, and double the answer to get the total volume of the solid.

Disk Method Vs Shell Method

Both the Disk method and the Shell method provide us with a way to calculate the volume of a solid created by revolving a two-dimensional shape around an axis. However, there are some key differences between the two methods that make them more suitable for some types of problems than others. Here are some of the main differences:

1. The shape of the solids: The Disk method is better suited for volumes of solids that are formed by revolving circular or semi-circular shapes around an axis, while the Shell method is better suited for volumes of solids that are formed by revolving regions bounded by two curves around an axis.

2. The slicing direction: The Disk method slices the shape into disks that are parallel to the axis of rotation, while the Shell method slices the shape into shells that are perpendicular to the axis of rotation.

3. The areas of integration: The Disk method integrates the area of the cross-section of the shape, which is a function of the distance from the axis of rotation, while the Shell method integrates the circumference of the cylindrical shells, which is a function of the height of the shells.

4. Ease of Integration: In certain problems, one method will have easier integrals to compute compared to the other method. For example, for a volume of a cone, it can be easier to use a shell method instead of a disk method.

When to use Disk Method and Shell Method?

To decide which method to use, we need to examine the shape and orientation of the solid, and determine which method best suits the problem at hand. As a general rule, Disk method should be used:

– When the shape is symmetrical, such as when revolving a circle around an axis.

– When the integrals for the area of the cross sectional area are more easily computed.

– When the axis of rotation is vertical (horizontal axis of the intersection).

– When a comparison of multiple figures such as solid vs donut.

On the other hand, the Shell method should be used when:

– The shape is not symmetrical, like when revolving a region bounded by two curves.

– When there is an easier method to find the length of the shell segments.

– When a comparison of disks and cans is needed.

Conclusion

The Disk Vs Shell method can both be used to calculate the volume of solids created by revolving a two-dimensional shape around an axis. Each method has its pros and cons, and the decision to use one method over the other should be based on the specific characteristics of the problem at hand. By understanding the applications and limitations of both methods, we can choose the one that is best suited for our needs, and tackle even the most complex volume calculation problems with ease.