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APPM 1360
Exam 2
Spring 2016
On the front of your bluebook, please write: a grading key, your name, student ID, your lecture number
and instructor. This exam is worth 100 points and has 5 questions on both sides of this paper.
• Submit this exam sheet with your bluebook. However, nothing on this exam sheet will be graded. Make
sure all of your work is in your bluebook.
• Show all work and simplify your answers! Answers with no justification will receive no points.
• Please begin each problem on a new page.
• No notes or papers, calculators, cell phones, or electronic devices are permitted.
1. (28 pts, 7 pts each) For each of the following parts, let f (x) = x2 + 2 and g(x) = 4 − x2 . For parts (a),
(b), and (c), set up, but do not evaluate, the integral needed to find the requested information.
(a) The volume of the solid obtained by rotating the region bounded between the graphs of f (x) and
g(x) about the horizontal line y = −3.
(b) The volume of the solid obtained by rotating the region bounded between the graphs of f (x) and
g(x) about the vertical line x = 2.
(c) The surface area obtained by rotating the graph of f (x) on the interval −2 ≤ x ≤ 2 about the
x-axis.
(d) Show that, on the interval −2 ≤ x ≤ 2, the arc length of f (x) is equal to the arc length of g(x).
(HINT: You do not actually need to evaluate any integrals here.)
2. (16 pts) Consider the trapezoidal region with uniform density ρ shown below.
(a) Find the moments My and Mx of the region using integration.
(b) Find the centroid of the region using your answers for part (a).
(c) Now draw a horizontal line at y = 2 to divide the trapezoid into a triangle and a rectangle. Locate
the centroids of the smaller regions and use additivity of moments to confirm your answer for part
(b).
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TURN OVER - More problems on the back!
3. (16 pts) Consider the differential equation
dy
x
= −x .
dx
ye
(a) Solve the differential equation.
(b) Find the solution to the differential equation that satisfies the initial condition y(0) = 2.
4. (20 pts) Consider the series
∞
X
n=1
n
X
2
an where an =
. Let the partial sum sn =
ai .
(n + 1)(n + 3)
i=1
(a) Write the partial fraction decomposition of an .
(b) Find a simple expression for sn .
(c) Is {sn } monotonic? Justify your answer.
(d) Is {sn } bounded? If so, find upper and lower bounds for sn .
(e) Does the given series converge? If so, what does it converge to?
5. (20 pts) For each of the following, please answer ‘True’ or ‘False’. Provide a brief justification of your
answer.
(a) The sequence {an }, given by an = ln(n!) − ln((n + 1)!) for n = 1, 2, 3, . . . converges.
∞
X
(b) If lim an = 0 then
an is convergent.
n→∞
(c) The series
n=1
∞
X
1 + 2n
n=2
(d) Let an = e
−n
3n
converges.
−1
and bn = tan
n. The series
∞
X
(an + bn ) diverges.
n=1
(e) Suppose you are slicing a ten inch long carrot really thin from the greens end (x = 0) to the tip of
the root (x = 10). If each slice has a circular cross section with area f (x) = π[r(x)]2 for each x
between 0 and 10, and we make our cuts at x1 , x2 , ..., xn−1 (so x0 = 0 and xn = 10) then a good
n
X
approximation for the volume of the carrot is
f (xi )xi .
i=1
Some Trigonometric identities
2 cos2 (x) = 1 + cos(2x)
2 sin2 (x) = 1 − cos(2x)
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2 (x) − sin2 (x)
Center of Mass Integrals
Z b
M=
ρ(f (x) − g(x)) dx
a
b
Z
ρx(f (x) − g(x)) dx
My =
a
Inverse Trigonometric Integral Identities
Z
du
√
= sin−1 (u/a) + C, u2 < a2
a2 − u2
Z
du
1
= tan−1 (u/a) + C
2
2
a +u
a
Z
du
1
√
= sec−1 |u/a| + C, u2 > a2
2
2
a
u u −a
Z
Mx =
a
x̄ =
b
1
ρ(f 2 (x) − g 2 (x)) dx
2
My
Mx
and ȳ =
M
M