__Category__: Integral Calculus, Analytic Geometry"Published in Newark, California, USA"

Find the volume generated by revolving about the x-axis and y-axis the areas bounded by the curves

__Solution__:

To illustrate the problem, it is better to sketch the graph of the three equations above using the principles of Analytic Geometry as follows

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From the figure above, the point of intersection between y = x

^{3}and y = 0 is (0, 0), the point of intersection between y = 0 and x = 2 is (2, 0), and the point of intersection between y = x

^{3}and x = 2 is (2, 8) by substituting x = 2 to y = x

^{3}.

Next, from the given three equations above, it is better to use a vertical strip at the area bounded by three curves and label further the figure as follows

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If you rotate the shaded area about the x-axis, the vertical strip becomes a disk as follows

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The volume of a disk formed by the rotation of a vertical strip about the x-axis is

Integrate on both sides of the equation to get the volume of a solid formed by the rotation of the area about the x-axis as follows

If you rotate the shaded area about the y-axis, the vertical strip becomes a cylindrical shell as follows

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The volume of a cylindrical shell formed by the rotation of a vertical strip about the y-axis is

The circumference of the base of the cylindrical shell is 2πx. x is the distance of the vertical strip to the axis of revolution which is the y-axis. y is the height of a cylindrical shell. dx is the thickness of a cylindrical shell. Since dx is a very small value, then the two radii of the cylindrical shell are almost the same which is x. The three dimensions of a thin rectangular box are 2πx, y, and dx. When you wrapped a thin rectangular box into a cylinder, then it becomes a cylindrical shell. Therefore, the volume of a cylindrical shell is

Integrate on both sides of the equation to get the volume of a solid formed by the rotation of the area about the y-axis as follows