George Lerakis explains how to find volume by the method of cylindrical shells. He starts with when someone has to use this method (0:20)
Brief theory (6:00)
Example (13:05)
Until now Lerakis continues we know how to find volume of a solid created when we rotate the graph of a function f(x) about the x-axis by the method of cylindrical disks where one has to find the definite integral of pi.[f(x)]^2 with respect x. But this method Lerakis continues has limitations on it's applications, because when a graph of a relation that is not a function from x to y is rotated around the x-axis then in some points you have to radius an inner one and an outer one which makes it difficult to apply the disks method. In some cases is even impossible because we can't solve the equation to find the formulas for the upper and lower part of the graph.
That's when we have to use the method of cylindrical shells. in this case Lerakis continues we take a parallelogram of thickness dy which we rotate around the x-axis or an axis parallel to x-axis it creates a cylindrical shell with volume 2.pi.y.xdy. Therefore in order to find the volume of the figure created when we rotate the graph around the x-axis we have to take the summation of all these cylindrical shells which means that we have to take the definite integral of 2.pi.x.ydy from a to b where a and b are the lower and upper y values of the graph that we rotate
Lerakis continues that something worth mentioning here is that the difference between the two methods is that in the disks method when rotating around the x-axis or axis parallel to the x-axis we take the integral with respect with dx while with the shells method we take the integral with respect to dy.
Finally Lerakis continues with an example, which is to find the volume of the solid created when we rotate the graph of the function x=2y^2-y^3 around the x-axis, where he explains why one has to use the method of cylindrical shells. He uses solving a third degree equation with some factoring to find the limits of the integral and the rule of integration that the integral of x^n is x^(n+1)/(n+1).
www.mathphysicseducation.com
[ Ссылка ]

свернуть