Estimating the Volume of Solid of Revolution by Shell Method and Washer Method?

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 d 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 the objects or shapes occupy space and physical dimensions. Then the shell method will help in measuring these objects by the shell methods.

Shell Method Formula

The volume by shells is obtained by rotating the region y = f(x) when rotated along the x-axis and y-axis in an interval of [a, b]. Consider we have a cylindrical shell having radius “r”, height “h”, then its area will be 2πrh.

So the formula for obtaining the volume for shell methods is as follow:

Besides using these formulae and doing complex calculations an individual may also calculate shell volume by using a volume by cylindrical shells calculator.

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 the 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 which 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.

Washer Method Formula

The washer Method for integration helps us to find the volume of solid of revolution in which the axis of rotation is not adjoint with the boundary of the plane. Also, the cross-section area is taken perpendicular to the axis of rotation for calculating it manually. 

In the manual process, the washer method uses the formula for solving the integral function under the integral, but with the help of the online washer method to find a volume calculator, you can perform it online without doing any brainstorming. 

But Beside to this, both methods uses the same formula’s given as follow:

Rotation along x-axis

abπ ( [ f(x)]2-[ g(x)]2) dx

Rotation along y-axis

abπ ( [ f(y)]2-[ g(y)]2) dy

Is the Shell Method the same as the Washer Method?

As we know about both concepts of the shell method and the washer method there is a main difference at the 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 are 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, the basic and important difference between the shell method and the washer method is that 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 method and washer method can be differentiated by the examples and 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, and their differences among them. It was confirmed in this article that the shell method is not as same as the washer method because the washer method is used between the curves whereas the shell method is used in the holes like disc holes.