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Peer Questions for Section 7.6, omitting the subsection pp. 401-403 on Hydrostatic
Pressure and Force
On Tues., Oct. 1, you will be discussing your answers in group. Given the nature of answers, and
the extraordinarily long length of this list of questions (many devoted to reviewing ideas from
Chapter 7), you will not be submitting your answers to me. Make sure to place emphasis on parts
1(f)–(g), 2(f)–(g), 5 and 6, as these are the ones from the new section.
1. (Parts (a)–(e) are review from earlier sections of Chapter 7.) Consider the region pictured.
Write formulas for
(a) the length of the curve y “ f pxq, a ď x ď b.
(b) the volume of the solid of revolution when rotated about the line
y “ ´1.
(c) the volume of the solid of revolution when rotated about the y-axis.
(d) the (lateral) surface area of the solid of revolution from the part (b).
(e) the area of the region that lies between y “ f pxq and y “ 1 ´ f pxq.
You may assume that the minimum value of f inside a ď x ď b is 2.
(f) the moments about the x- and y-axis, assuming constant mass density ρ.
(g) the mass of the region, assuming constant mass density ρ.
2. (This problem reviews material prior to Section 7.6.) Suppose our region in Problem 1 is that
?
bounded by y “ 1 ` x ´ 1, 1 ď x ď 4, and the x-axis. What changes can be made to the
formulas you wrote for Problem 1 to reflect the specifics of this region?
3. (This problem reviews material prior to Section 7.6.) You probably did Problem 1(c) using
cylindrical shells. Set it up via the method of disks/washers, when the region being rotated
is that given in Problem 2.
4. (This problem reviews material prior to Section 7.6.) Think up, and attempt to answer, altered
versions of the questions in Problem 1 that pose additional challenges.
5. (A question related to material in Section 7.6.) What is the difference between weight and
mass?
6. (A question related to material in Section 7.6.)
paradigm:
Some work problems fit the following
• you have a point mass (or a mass that essentially could be shrunk down to a point, like
one at the end of a spring).
• you have a formula for the force acting on that point mass, a formula f pxq which changes
with position.
• you want to calculate the work in moving the point mass from position x “ a to x “ b.
These are very easy problems to set up, as the formula for the work is given by
żb
f pxq dx.
a
Examples 1 and 2 in Section 7.6 are of this type.
Examples 3 and 4 of this section do not fit this paradigm, however. Looking at the details
of Example 3, we see that p2∆xq (which becomes 2dx in the integral) represents the force; in
Example 4, the force (written as 1568πp10 ´ xq2 ∆x) contains ∆x again (so, in the resulting
integral, you must think of dx as part of the force). Can you come up with a description of
the paradigm for these problems or, perhaps even more useful, a description for how they
are solved?
7. Identify one item (a concept, a step in an example, a statement, etc.) from this reading
assignment you found difficult or confusing.