What is a Solid of Revolution - 1 Consider the area under the graph of y = 0.5x from x = 0 to x = 1:
What is a Solid of Revolution - 2 If the shaded area is now rotated about the x-axis, then a three- dimensional solid (called Solid of Revolution) will be formed: http://chuwm2.tripod.com/revolution/Pictures from What will it look like?
What is a Solid of Revolution - 3 Actually, if the shaded triangle is regarded as made up of straight lines perpendicular to the x-axis, then each of them will give a circular plate when rotated about the x-axis. The collection of all such plates then pile up to form the solid of revolution, which is a cone in this case.
How is it calculated - 1 Consider the solid of revolution formed by the graph of y = x 2 from x = 0 to x = 2: What will it look like? http://www.worldofgramophones.com/ victor-victrola-gramophone-II.jpg
How is it calculated - 2 Just like the area under a continuous curve can be approximated by a series of narrow rectangles, the volume of a solid of revolution can be approximated by a series of thin circular discs: we could improve our accuracy by using a larger and larger number of circular discs, making them thinner and thinner
How is it calculated - 3 x x x As n tends to infinity, It means the discs get thinner and thinner. And it becomes a better and better approximation. As n tends to infinity, It means the discs get thinner and thinner. And it becomes a better and better approximation. It can be replaced by an integral
Volume of Revolution Formula The volume of revolution about the x-axis between x = a and x = b, as, is : This formula you do need to know Think of is as the um of lots of circles … where area of circle = r 2
How could we find the volume of the cone? One way would be to cut it into a series of disks (flat circular cylinders) and add their volumes. The volume of each disk is: In this case: r= the y value of the function thickness = a small change in x = dx Example of a disk
The volume of each flat cylinder (disk) is: If we add the volumes, we get:
Example 1 Consider the area under the graph of y = 0.5x from x = 0 to x = 1: What is the volume of revolution about the x-axis? Integrating and substituting gives: 0.5 1
Example 2 between x = 1 and x = 4 What is the volume of revolution about the x-axis Integrating gives: for