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1. Hal has just finished making a ring, which is given by the rotation of the region between
-1
j(x) = 2 - 5x + 4'
+ = 1.2, x = 3.3, and y = 0 about the line r = 0.5.
(a) If r and y are both measured in centimeters, what is the volume of the ring? Calculate the
value of the integral you set up by hand, showing your solution step-by-step.
(b) The ring's density is given by p(r) grams/ centimeter, where r is the distance in centimeters
from the central axis of the ring. Write an integral expressing the total mass of the ring. (Note
that you do not need to calculate this integral.)
(c) Hal decides the ring isn't quite what he wants, and decides instead to make a ring by rotating
the same region around the line y = -0.9. Write an integral expression that gives the total
volume of this new ring.
(d) Hal wants to put a colorful piece of green tape around the ring, as shown. Write an integral
expression that gives the length of the piece of tape.
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