Unformatted text preview: Workshop 2
1. Start with the region A in the ﬁrst quadrant enclosed by the x-axis and the parabola
y = 2x(2 − x). Then obtain solids of revolution S1 , S2 , and S3 by revolving A about the lines
y = 4, y = −2, and x = 4 respectively. All three solids are (unusual) “doughnuts” which are 8
units across, whose hole is 4 units across, and whose height is 2 units. Sketch them.
a) Which do you expect to have larger volume, S1 or S2 ? Compute their volumes exactly and
check your guess.
b) Compute the volume of S3 . (It may be harder to guess in advance how S3 compares in
volume to S2 and S1 .) 2.Electrons repel each other with a force which is inversely proportional to the square of the
distance between them; call the proportionality constant k in the units to be used. Suppose
one electron is ﬁxed at x = 0 on the x-axis.
a) Find the work done in moving a second electron along the x-axis from the point x = 10 to
the point x = 1.
b) Find the work done in moving the second electron along the x-axis from the point x = M
to the point x = 1. c) What happens to your answer in b) (which should depend on M ) as
M → ∞? 3. A homogeneous liquid whose density is 300 kg/m3 ﬁlls three buried containers. The containers, drawn below, are each 10 meters tall. The top of each container is at ground level.
All three containers have the same volume. The middle container is a cylinder, and the other
two are circular cones. Which container needs the least amount of work to empty (that is, to
pump the liquid to ground level)? Which container needs the most work to empty? Justify
your assertions by computing the work necessary in each case. You may also discuss why your
answer is correct!
ground level 10 m. 3 m. ...
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