4 questions, On midterm 1, astronomy students from school A recently observed an

4 questions,
On midterm 1, astronomy students from school A recently observed an unidentified object moving straight down toward them. The object descended quickly, stopping and hovering in the sky above the Laboratories.
The object is difficult to see clearly, even as it hovers apparently motionless in the sky. Despite observing the object with their
best telescopes and having their most sensitive radio receivers trained on it, they are only able to get a few scattered data
points about the shape of the object, and some scrambled transmissions.
The students are able to get five data points about the top and bottom surfaces of the object, to help them approximate its
shape. They represent these data points as (x, y) coordinates, as follows.
(in uploads)
The blue
points and red points represent the top and bottom surfaces, respectively.
These quantities are all in units of tens of metres. For example, the point (2, 3) on the top surface is 20m to the right and
30m above the origin of this coordinate system.
Other than these data points, the astronomy students are quite unsure of the actual shape of the object. Their best idea
is to use their data to approximate the shape of the region enclosed by the two surfaces, then rotate that region around a
coordinate axis to try to approximate the object as a solid of revolution.
(plot of data points in uploads)
Unfortunately, it’s been a while since these students took math, and they’re rusty. They try to refresh their
memories about solids of revolution, and fall back into making common errors while working on the following problem.
Problem
Consider the region R, in the first quadrant, bounded between the curves y= √x, Find the volume of the solid of revolution obtained by rotating R around the x-axis.
y= x− 2, and the x-axis
They are able to find the intersection point of the two curves correctly:
√x = x− 2 = ⇒ x = 4 and therefore y = 2, so the intersection point is (4, 2).
But things do not go well from that point onward.
a)Jimmy is the first student to propose a solution.
Jimmy’s attempt
The √x curve is above the x− 2 line between 0 and 4. The region starts at x = 0 and ends at x = 4.
That means its volume is:
V= ∫4
0 [π (√x)2
− π(x− 2)2]dx
Jimmy is wrong. In at most a few sentences (and some pictures, if you like), explain the error(s) he is making.
b)Next, Suzie tries the question.
Suzie’s attempt
I can see that this integral should be in two parts, changing at x = 2. The cross-sections of the first
part are circles with radius √x, and the cross-sections of the second part are washers with inner radius
x− 2 and outer radius √x, respectively. That means the volume is:
V= ∫2
0
(π√x)2 dx + ∫4
2
π (√x− (x− 2))2 dx
Suzie is also wrong. In at most a few sentences (and some pictures, if you like), explain the error(s) she is making.
c)Show these students who’s boss by doing a scarier looking version of the same problem.
Let R be the same region described in the problem above (remember, R is in the first quadrant only, above the
x-axis). Find a sum of integrals representing the volume of the solid of revolution obtained by rotating R around
the the line y=−136.
Explain how you get your final answer, but you do not have to evaluate any integrals.
Having proven their skills, school B students are called in to help the student A astronomers understand the mysterious
object, which they hope is an alien spacecraft. We have the data from the table, but we can’t know its true shape for sure. So,
the school B students do what they know best: approximate the shape with rectangles.
Over each of the four intervals [2, 4], [4, 6], [6, 8], and [8, 10], use the y-values from the two surfaces at the left endpoint of
the interval as the top and bottom of a rectangle. For example, over the interval [4, 6], the rectangle extends from 1 to 5.5.
Doing this for all four intervals produces a region S, consisting of four side-by-side rectangles.
2. School B start by doing the simplest things they can with the region S.
(a) (1 point) Draw a picture of S, and find the area of S.
(You may draw on the plot or print a copy of it to draw on. No integration required for the area…it’s just rectangles.)
(b) (1 point) Our first attempt at approximating the mysterious object is as the solid of revolution formed by rotating
S around the x-axis. Find the volume of this solid.
The school A is impressed. However, they note that the solid from Q2 doesn’t look much like the cool flying saucer
they hoped they had discovered, and you agree.
3. (2 points) Find the volume of the solid of revolution obtained by rotating the region S around the y-axis. (This looks
much more like a spaceship!) Be sure to explain your process for doing this computation in detail.
Eventually, they are able to see the precise outline of the re-
gion U they were approximating with S earlier. See the im-
age to the right.
They find that U extends all the way to the y-axis with two
horizontal lines, as shown. They also find that y= g(x) is
a straight line joining (2, 3) to (4, 6), and that the bottom
surface is given by the graph of the function
f (x) = 2/ln(9) ln(x− 1) (graph in upload) for 2 ≤ x ≤ 10.
The curve x = h(y), for 2 ≤ y ≤ 6, forming part of the top surface is an alien curve that school B students do not recognize.
The can only unscramble part of one signal from the ship, which says:
tra! I’m back … kill … all … human … And by the way, ∫6
2
[h(y)]2 dy=
…
The students are a bit worried about the first part, but also frustrated that they couldn’t unscramble the RHS of the equation.
They decide to call the missing value of the integral in the message A.
4. (3 points) Using the information they have gathered above, find the volume of the ship, which is the solid obtained by
rotating the image in the picture about the y-axis.
When simplified, the final answer should involve the constant A, a logarithm, a π, and some integers. It should not
contain integrals. Be sure to lay out your work clearly, and explain your steps in words and/or pictures.

. 4 questions,
On midterm 1, astronomy students from school A recently observed an on .

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