Mathematically the shell method is defined as the process of calculating or finding the volume of the solid of the revolution. When the integration is defined as along an axis that is perpendicular to the axis of revolution. In other things, the washer method is defined as the integration which is used to find the volume of a shape. We used the washer method to find the volume of the solids of the revolution.
It describes the function that takes place on an interval (x,y) and then rotated on a point x or y-axis. There would be a complete study of the shell method and the washer method. The shell method and the washer method integration can be understood by this article. The shell method integration and the washer method integration can be explored and also differentiates the differences between the shell method integration and the washer method integration.
What is the Shell Method?
In calculus, the shell method is explained as the method which helps in finding the volumes of the shape. Particularly in the process of finding the volumes that decompose a solid of revolution into the cylindrical shells.
As the shell method’s name indicates, the shell method is a shell integration method because it uses cylindrical shells. Solely the shell method is operated as when the integration along the axis is perpendicular to the axis of revolution, then there is a need of using the method in which the calculation of the volume of a solid of revolution is required.
In simple words, it can be said that the process of shell method integration will help in calculating the volume of revolution by summing the volumes of thin cylindrical shells in a limit. As we know that all objects or shapes occupy space and physical dimensions. Then the shell method will help in measuring these objects by the shell methods.
Related: Also learn more about how to find the volume of solid of revolution?
How to Calculate Volume By Shell Method?
The shell method in integration is basically used for finding the volume of solid of revolution. It is also known as the volume of cylindrical shells. Shell method is considered as the contrast method of the solid of the revolution in which cross-sectional area is taken parallel to the axis of rotation to find the volume of solid of revolution.
For finding the volume by shell method, one may need to follow these given formulas:
If the body is rotating along the x-axis in an interval of [a,b], then the volume by shell method formula is as follows:
Similarly, If the body is rotating along the y-axis in an interval of [a,b], then the volume by shell method formula is as follows:
Thus these are some easy formulas to apply to a function of any rigid body. Whose volume by shell method has required to calculate. Finding the volume of solid of revolution by shell method is no doubt a lengthy and complex process. But if you understand the whole idea it will definitely become an interesting and wonderful concept.
Students can also concern with the online cylindrical shell method calculator which helps them to solve the complex queries of finding the volume of solid of revolution by shell integration method.
What is the Washer Method?
In mathematics or calculus, the washer method is known as the process which helps in finding the volume of the objects of revolution. The basis of the x-axis or y-axis, in the cross-sections that look like washers, is known as the washer method because the thin or horizontal slice from the spherical shape on the left is rotated around the y-axis. It is the method that is also used to find the volume of solids of revolution.
For example, the solid of revolution acts as a function that takes on the interval ( x, y ) and then rotates on the axis or known as rotate at some point. Unusually, the washer method helps in finding out the volume of the solid even when it has two disks. It leads to the two-disc method. Washer method integration is also used in finding the solid which has a rectangle that sweeps out and is similar to the hole in the middle of the CD or any hole.
How to Calculate Volume By Washer Method?
The washer method in integration is basically used for finding the volume of solid of revolution. It is also known as the volume by ring method. It is basically used to find the solid of the revolution in which the axis of rotation is not attached to the boundary value of the plane. Moreover, the cross-sectional area is always taken perpendicular to the axis of the rotations in the ring/washer method.
For finding the volume by ring/washer method, one may need to follow these given formulas:
If the curved body is rotating between the functions of f(x) and g(x) along the x-axis in an interval of [a,b], then the volume by ring/washer method formula is as follows:
Similarly, If the curved body is rotating between the functions of f(x) and g(x) along the y-axis in an interval of [a,b], then the volume by ring/washer method formula is as follows:
Thus these are some easy formulas to apply to a function of any curved surface body. Whose volume by ring/washer method has required to calculate. Finding the volume of solid of revolution by ring/washer method is no doubt a lengthy and complex process. But if you understand the whole idea it will definitely become an interesting and wonderful concept.
Similar to the shell method integration, there is also a washer volume calculator available online for solving and understanding the volume of the solid of the revolution of the curved body. These online tools will surely helpful for students in their learning and solving queries related to a stepwise solution.
Is the shell method the same as the washer method?
We know concepts of shell method and the washer method there is the main difference at concerning point. The washer method and the shell method in calculus will explain the orientations to the axis of rotations. The washer method and the shell method is used when the derivatives of x rotate around the x-axis.
Whereas the shell method is used when the derivatives of y rotate around the x-axis. In simple words, there is a huge difference between the shell and the washer methods. The washer method is used between the two curves.
Conclusion
In this article, there is a complete explanation of the shell method and the washer method. The shell and washer methods can be differentiated on the basis of concepts of their uses.
The complete study of this article will help in knowing about the shell method and washer method. Also, we learned about the differences among them. It has confirmed in this article that the shell method is not as same as the washer method. Because the washer method has used between the curves. Whereas the shell method is used in the holes like disk holes.
So we hoped you may like this great article. You may also learn more about derivatives and its daily life applications offered by emu articles. So keep attached to us for happy learning.
