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

The shell method in mathematics is the process used to calculate or determine the volume of a solid of revolution. It involves integrating along an axis that is perpendicular to the axis of revolution. Similarly, the washer method is an integration technique used to find the volume of a shape.

These methods are employed to find the volume of solids that are created by rotating a function around an axis, such as the x or y-axis. A thorough understanding of the shell method and the washer method is essential for mathematical studies.

This article provides insight into both the shell method integration and the washer method integration. It also highlights the differences between these two integration techniques.

What is the Shell Method?

In calculus, the shell method is used to determine the volumes of shapes, particularly in decomposing a solid of revolution into cylindrical shells.

As its name suggests, the shell method involves cylindrical shells. It is employed when the integration along an axis is perpendicular to the axis of revolution, allowing for the calculation of the volume of a solid of revolution.

In essence, the shell method integration calculates the volume of revolution by summing the volumes of thin cylindrical shells within a limit. Since all objects or shapes occupy space and have physical dimensions, the shell method aids in measuring these objects.

Shell Method Formula

The volume using shells is found by rotating the region y = f(x) along the x-axis and y-axis within an interval [a, b]. If we have a cylindrical shell with radius “r” and height “h,” its area will be 2πrh.

Therefore, the formula for calculating the volume using the shell method is as follows:

To estimate the volume of a solid of revolution using the shell method and washer method, an individual may use a calculator designed for calculating volumes using cylindrical shells.

What is the Washer Method?

In mathematics or calculus, the washer method is a technique used to determine the volume of objects formed by revolution. It derives its name from the cross-sections resembling washers that result from rotating a thin or horizontal slice from a shape around the y-axis.

This method is commonly employed to find the volume of solids of revolution. For instance, a solid of revolution can be visualized as a function that spans an interval (x, y) and rotates around an axis or point.

Moreover, the washer method facilitates the calculation of the solid’s volume even when it includes two disks, which is known as the two-disc method. This integration technique is also applicable in scenarios where the solid resembles a rectangle sweeping out and resembling the hole in the center of a CD or similar object.

Washer Method Formula

The washer method for integration is utilized to determine the volume of a solid of revolution where the axis of rotation is not contiguous with the boundary of the plane. This method involves calculating the cross-sectional area perpendicular to the axis of rotation manually.

In the manual approach, the washer method employs a formula to solve the integral function under the integral sign. However, using an online washer method volume calculator simplifies this process, eliminating the need for complex calculations.

Both methods, manual and online, utilize the same formulas, as shown below:

Rotation along the x-axis:

π∫ab​([f(x)]2−[g(x)]2)dx

Rotation along the y-axis:

π∫ab​([f(y)]2−[g(y)]2)dy

Is the Shell Method the same as the Washer Method?

In comparing the shell method and the washer method, the main distinction lies in their orientation to the axis of rotation. The washer method and the shell method in calculus explain how derivatives of x rotate around the x-axis, while the shell method is used for the derivatives of y rotating around the x-axis.

Put simply, the key difference between the shell method and the washer method is that the washer method is employed between two curves.

Conclusion

This article provides a comprehensive explanation of the shell method and the washer method, highlighting their differences through examples and conceptual uses.

A thorough understanding of this article will clarify the distinctions between the shell method and washer method. It emphasizes that the shell method differs from the washer method, as the washer method is applied between curves, while the shell method is used for voids resembling disc holes.