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Find the volume generated by revolving about the x-axis and y-axis of the area bounded by the following curves
Solution:
To illustrate the problem, it is better to sketch the graph of the two equations above using the principles of Analytic Geometry as follows
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| Photo by Math Principles in Everyday Life |
From the figure above, there are two points of intersection between the two curves. Solve the systems of two equations two unknowns in order to get the coordinates of the points of intersection as follows
we have,
Equate each factor to zero and solve for the value of x as follows
If
If
Therefore, their points of intersection are (0, 0) and (2, 4).
Next, from the given two equations above, it is better to use a vertical strip at the area bounded by two curves and label further the figure as follows
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| Photo by Math Principles in Everyday Life |
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| Photo by Math Principles in Everyday Life |
The volume of a ring 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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| Photo by Math Principles in Everyday Life |
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
