Volume of a solid of revolution: method of cylindrical shells

In this post we are going to compute the volume of a solid rotating around the y-axis by utilizing the method of cylindrical shells.

Let's consider a function f. positive over an interval [a,b]. Let R be the region bounded by the curve of the function, two verticals x = a and x = b and the x-axis..

A solid of revolution is obtained by revolving the region around the y-axis. Our goal is to determine the volume of this solid. We are going to use the method of cylindrical shells for that purpose.

Method.

Let's divide the region R in small rectangles of width dx and consider one rectangle.

Let's revolve the rectangle around the y-axis. We obtain a cylindrical shell



Let's calculate the volume of this cylindrical shell. The volume of this cylindrical shell is obtained by multiplying the volume of the outer cylinder by the thickness of the cylindrical shell. We can write:

Volume cylindrical shell = Volume outer shell * thickness.

The volume of a cylinder is equal to the surface of the base by the height. Therefore the volume of the outer shell can be written as:

Volume of outer shell = Area of base * height. The base is a circle. Its area is 2𝜋r. Then
Volume outer shell  = 2𝜋rh .

Let's substitute in Volume cylindrical shell

Volume cylindrical shell = 2𝜋rhdx (dx is the thickness)

The radius r is variable. We call it x. The height is a function of the radius. We represent it by f(x)

V cylindrical shell =  2𝜋xf(x)dx
Adding the volume of all these small cylindrical shells is going to give the volume of the entire solid of revolution. This is equivalent to integrate the volume over the interval  [a,b]. Therefore the volume of the solid of revolution is given by:

Example

Use the cylindrical shells to find the volume generated when the region bounded by the curves is revolved about  the y-axis.

y = 1/x  y = 0 x = 1 x = 3

Solution 

The volume of a cylindrical shell generated by a region around the y-axis is given by 

Practice

Find the volume generated by the lines y = 2x-1 y = -2x + 3 and 
x  = 2 about the y-axis.

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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]

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