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Solid of Revolution. Revolution about x-axis. 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.
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Solid of Revolution Revolution about x-axis
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: What will it look like? Pictures from http://chuwm2.tripod.com/revolution/
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.
Finding Volume http://clem.mscd.edu/~talmanl/HTML/VolumeOfRevolution.html
What will it look like? http://www.worldofgramophones.com/ victor-victrola-gramophone-II.jpg How is it calculated - 1 Consider the solid of revolution formed by the graph of y = x2 from x = 0 to x = 2:
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
As n tends to infinity, It means the discs get thinner and thinner. And it becomes a better and better approximation. 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. It can be replaced by an integral
Think of is as the um of lots of circles … where area of circle = r2 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
Example of a disk The volume of each disk is: 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. In this case: r= the y value of the function thickness = a small change in x =dx
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? 1 0.5 Integrating and substituting gives:
Example 2 What is the volume of revolution about the x-axis between x = 1 and x = 4 for Integrating gives:
What would be these Solids of Revolution about the x-axis? y y x x Sphere Torus
What would be these Solids of Revolution about the x-axis? y y x x Sphere Torus
