| Theorem of Pappus |
The Theorem of Pappus states that when a region R is rotated about a line l, the volume of the solid generated is equal to the product of the area of R and the distance the centroid of the region has traveled in one full rotation. The centroid of a region is essentially the one point on which the region should "balance." The centroid of a rectangle with vertices (0,0), (x,0), (0,y), and (x,y) is the point (x/2,y/2), for example, but finding the centroid of a non-rectangular region is a little bit trickier. Part of this week's problem will require you to come up with a unique way of locating the centroid of a semicircle.
Consider the figure below, a rectangle topped by a semicircle.
Use the Theorem of Pappus to:
1) Find the centroid of the semicircle and use it to find the volume of the solid generated when just the semicircle is rotated about l.
2) Find the volume of the solid generated when just the rectangle is rotated about l.
3) Find the distance from the centroid of the region R to l. |
Problem Moderated by: Sasha |
| Problem Solution |
First, we must find the centroid of the semicircle. Consider: a semicircle of radius 3π/4, when rotated about its diameter, generates a solid with a volume of (4/3)π(3π/4)3. The cross-sectional area of this solid is (1/2)π(3π/4)2, and the distance traveled by the centroid is 2πk, where k is the distance from l to the centroid. (We can take for granted that the centroid lies along the line perpendicular to l which bisects the area of the circle.) We find that k = 1, conveniently. Therefore the distance from l to the centroid of the semicircle in the actual figure is 3, and the volume swept out is
(1/2)π(3π/4)2*2π*3 = 27π4/16
The area of the rectangle is 2(3π/2) = 3π
The distance traveled by the centroid is 2π. Therefore, the volume swept out is 6π2, and the total volume is
27π4/16 + 6π2
The area of the entire region is 3π + 9π3/32
Therefore, (3π + 9π3/32)(2π*k) = 27π4/16 + 6π2
The value of k is (9π2 + 32)/(3π2 + 32) |
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