MATH128 { Spring 2018
Assignment 2
Due 31 May 2018 at 17:00
This assignment has to be uploaded to Crowdmark by the due date. Since no extensions will be
granted make sure you can access the system in good time. The Crowdmark link for submission will
be sent approximately one week in advance of the deadline. In case of malfunctions the solutions
can be sent by email to . However, they must be received before the
due date and time.
There are 56 points available on the questions plus an additional 5 for presentation. You are
expected to show your full workings for every question. Make sure you are justifying all integrals
which are not in the \standard integral" list available on LEARN.
1. Improper Integrals. Determine whether the following integrals converge. If they do, nd
their value.
(a) (2 points)
Q1 Total: 10 points
2. Area between curves.
(a) (4 points) Using integration, nd the area of the circle of radius 2 that has an ellipse
removed with semi-major axes of length 2 and semi-minor axis of length 1=2, see the gure
below.
(Hint: An ellipse satis es the equation (x=a)2 + (y=b)2 = 1 for some constants a;b > 0.)
(b) (3 points) Let f(x) = 3e x+logx and g(x) = logx. Determine whether the area bounded
between y = f(x), y = g(x), for x 0, is nite. If so, calculate its value. See the gure
below.
Q2 Total: 7 points
3. Volume of revolution.
(a) (5 points) Find the volume of an ellipsoid that is obtained by rotating the ellipse given by
(x=2)2 + (5y)2 = 1 about the x-axis.
(b) (6 points) Let f(x) = 1, g(x) = 1=8x2 + 1=2, a = 2, b = 2. Graph f and g from a to b
and indicate the area bounded between y = f(x), y = g(x), x = a, and x = b. Calculate
the volume of the solid obtained by rotating this region about the x-axes.
Q3 Total: 11 points
4. (8 points) Cylindrical Shells. A torus is a shape resembling a doughnut. It is a cylinder of
radius r bent in such a way that the ends meet and that the central axis forms a circle with
radius R, see the gure below. Note that the outer diameter of the torus is 2(R + r) and the
inner diameter is 2(R r). Using the cylindrical shell method calculate the volume of a torus
with tube diameter r and bend diameter R.
Page 2
5. (8 points) Surface Area of Bodies of Revolution. Using the integration methods for
surface area of volumes of revolution, calculate the surface area of a sphere of radius 2 with a
hole of radius 1=2, see gure below.
6. Sequences.
(a) (4 points) De ne a sequence (an)n as follows: Let a0 = 1, a1 = 2, a2 = 0, and ak+3 =
ak+2 ak+1 +ak for all k 0. Write down the rst 8 terms of the sequence and determine
whether it converges.
(b) (4 points) Use the Squeeze Lemma to show that the sequence (an)n given by
ak = ( 1)k k!(k + 1)!
converges.
(c) (4 points) Let (an)n be any sequence of numbers such that an < 0. Use the Monotone
Convergence Theorem to show that the sequence (bn)n given by bk = maxfa0;a1; ;akg
is convergent.
Q6 Total: 12 points