PURE MATHEMATICS - Integration

 

Volumes of Revolution

 

 

method

rotation x-axis

rotation y-axis

 

 

 

Method

 

A volume(rotated around the x-axis)is calculated by first considering a particular value of a function, y1, up from a value of x at x1 .

 

The line x1y1 may be considered as the 'radius' of the solid at that particular value of x.

If you were to square the y-value and multiply it by pi, then a cross-sectional area would be created.

 

Making a solid of revolution is simply the method of summing all the cross-sectional areas along the x-axis between two values of x.

 

(compare: area of a cylinder = cross-sectional area x length)

The method for solids rotated around the y-axis is similar.

 

Rotation around the x-axis

 

 

vol of revolution diagram#1

 

 

The volume Vxof a curve y=f(x) rotated around the x-axis between the values of x of a and b, is given by:

 

 

 

Example

 

What is the volume V of the cone swept out by the line y=2x rotated about the x-axis between x=0 and x=5 ?

 

 

vol of revolution problem#1

 

 

using   

 

and substituting                                  

 

      therefore              

 

                                               

 

     

 

 

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Rotation around the y-axis

 

 

vol revolution y-axis

 

 

The volume Vyof a curve y=f(x) rotated around the x-axis between the values of y of c and d, is given by:

 

 

Example

 

What is the volume V of the 'frustrum'(cone with smaller cone-shape removed) produced when the line y=2x/3 is rotated around the y-axis, when the centres of the upper and lower areas of the frustrum are at 0,7 and 0,3 ?

 

 

solids of revolution problem#2

 

 

 

 

 

                          

 

 

 

 

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