Volume of a solid of revolution: method of washer

According to the Merriam-Webster dictionary a washer is a flat ring or a perforated plate used in joints or assemblies to ensure tightness, prevent linkage or release friction.



The automobile mechanic technician use this type of ring to relate two pieces in fixing a car..

Washer method to compute the volume of a solid of revolution 

Let's consider two continuous and positive functions f and g over an interval [a,b]. Let R be the region bounded by the curves of these functions.



Let's have the shaded region revolve around the x-axis.


This is the solid we obtain. It has the shape of a hollow pot. In order to find its volume we have to integrate the cross-section, which is a washer or a flat ring as defined previously. The cross-section is shaded in the figure above.

The area of the cross-section is the difference between the area the of the circle of radius f(x) and the circle of radius g(x).

The volume of the solid of revolution is found by integrating the area:

Example

Find the volume generated when the region between the graphs f(x) = x² + 1 and g(x) = x over the interval [0,3] is revolved around the x-axis.


Solution


When the green region revolves around the x-axis it creates a solid of revolution like this:



The cross-section is a ring or washer. The volume of the solid it generates is given by:

Exercise

Find the volume generated when the region between the graphs f(x) = 2x² + 3 and g(x) = 2x over the interval [0,2]

Interested in Calculus tutoring visit New Direction Education Services for contact information
Interested in learning Calculus online visit Center for Integral Development