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Find the volume of the solid generated by revolving about y = -1 for the area bounded by the given 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
Equate
we have,
Therefore, their points of intersection are (-2, 5) and (2, 5).
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 |
If you rotate the shaded area about y = -1, the vertical strip becomes a ring as follows
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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
Since the axis of rotation is NOT the x-axis as stated in the given problem, then we have to get the outer radius and inner radius first. The axis of rotation is one unit below the x-axis which is y = -1. The outer radius and inner radius of a disk will be increased by one unit as follows
Substitute the values of the variables to the above equation, we have
Simplify the fractions by getting their Least Common Denominator (LCD) and the volume of the solid of revolution along y = -1 is
