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This article explores the concept of solids of revolution, specifically focusing on those formed by rotating areas under curves about the x-axis. By examining the area under the graph of y = 0.5x from x = 0 to x = 1, we visualize the resulting three-dimensional solid. We delve into volume calculation methods by approximating the volume using thin circular discs and employing integrals for enhanced accuracy. The principles along with practical examples illustrate how to effectively compute the volumes of various solids formed by revolution.
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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
