Washer and Disk Method: Understanding the Basics

When it comes to calculus, there are numerous methods to calculate the volume of a 3-dimensional object. Two of the most commonly used methods are the Washer Method and the Disk Method. These methods are used mainly to calculate the volume of a solid generated by revolving a two-dimensional shape around a line of axis. This article discusses the differences between these two methods and sheds light on how to perform each of these methods.

Washer Method:

The Washer Method, also known as the ring method, is used when the object is generated by revolving a function around an axis. Consider a function f(x) and a line L. The Washer Method is used when the solid is generated by revolve this function around the line L. The formula to calculate the volume of the solid using the washer method is given as:

V = π∫_a^b (R²-r²) dx

where R is the external radius, r is the internal radius, and dx is the thickness of the washers. “a” and “b” represent the limits of integration.

To calculate the external radius, R, we need to first find the distance of the point on the function from the line L. This is given by the equation (x-L). The external radius R is the distance from the line L to the point f(x) on the function. Similarly, to calculate the internal radius, r, we need the distance between the point on the function and the line L+ dx. This is given by the equation (x-L-dx). The internal radius, r, is the distance from the line L+dx and the point f(x+dx) on the function.

Disk Method:

The Disk Method is used to calculate the volume of a solid generated by revolving a curve around an axis line. Consider a curve y=f(x) and a line L. The Disk Method is used when the solid is generated by revolving this curve around the line L. The formula to calculate the volume of the solid using the disk method is given as:

V= π∫_a^b (f(x)² dx)

where “a” and “b” represent the limits of integration.

To calculate the volume using the Disk Method, solve for y in terms of x and rotate it around the axis of integration. The thickness of each disk is given by dx.

Differences between the Washer and Disk Methods:

Both the Washer and Disk Methods can be used to calculate the volume of a solid. However, the choice of method depends on the object you are working with. In the Washer method, the object is generated by revolving a function around an axis, while in the Disk method, the object is generated by revolving a curve around an axis. In the Washer Method, you need to find the internal and external radii, whereas in the Disk Method, you only need to focus on the function f(x) and thickness dx.

FAQs:

1. Is the washer method the same as the shell method?

No, the Washer Method is different from the Shell Method. The Shell Method is used to find the volume of a solid generated by revolving around an axis using cylindrical shells.

2. Which method should I use to calculate the volume of a sphere?

The Disk Method can be used to calculate the volume of a sphere. Consider a circle with radius r, and rotate the circle around the diameter to get a sphere.

3. Can I use the Washer Method if the axis of revolution is not perpendicular to the base?

Yes, you can use the Washer Method even if the axis of revolution is not perpendicular to the base. In this case, you need to find the radius of each washer at different points.

In Conclusion:

Both the Washer and Disk Methods are commonly used to calculate the volume of a solid. The method to use depends on the object being worked with. The Washer Method is used when the solid is generated by revolving a function around an axis while the Disk Method is used when the solid is generated by revolving a curve around an axis. When calculating the volume of an object, carefully analyze the object and decide which method will work best.