MAT 137 - ASSIGNMENT #8
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MAT 137 - ASSIGNMENT# 8
DUE MARCH 5
Instructions
Please submit solutions to the questions listed below on Crowdmark. Your assignment is due before 11.59pm on March 5. You should have received an email from Crowdmark containing the assignment questions and a link you can use to submit your solutions. You will need to submit a separate file for each question, so it is easiest to write up your solution to each question on its own paper. Please see the “Crowdmark How-To” document on the course page for tips about simplifying the submission process.
Your solutions should be neat and legible (preferably in black or blue ink). The written assign- ment is worth 10 points, and we will grade five (5) questions.
Technical issues will not be accepted as a reason for failing to submit on time, so please leave yourself ample time to submit your HW before the deadline.
Assigned Questions
(1) Compute the following integrals.
(a) dr.
r2 + 2r + 2
r + 5
(2) Compute the following integrals.
(a) t arctan(t2 )dt.
(b) Let f : R - R be a differentiable function so that f\ is differentiable and f\\ is continuous.
1
uf\\ (u)du
(3) Determine if the integral
o
r2 e北 dr converges or diverges. If it
一o
converges, find the value.
(4) We will determine if
o
e一^l北 ldr converges or diverges.
一o
(a) Compute te一tdt. (No it’s not a typo.)
(b) Compute e一^北 dr. (Hint: (a) will help after making a substitution.)
o b
o
(d) Use (b),(c) determine if e一^l北 ldr converges or diverges. If it converges, find the
一o
value.
(5) Let f : R - R be an odd continuous function. It would be tempting to claim that
b
f (x)dx = 0 for all b > 0.
一b
f (x)dx = 0
一o
o b
However, f (x)dx 
lim f (x)dx so we can’t make this claim in general.
o b→o 一b
(a) Use the definition of the improper integral to show that
that the claim is not true!
o
(b) However, now suppose that f (x)dx converges.
o
x3 dx diverges. This shows
一o
一o
Prove that in this case we must have
o
f (x)dx = 0.
一o
o − kz2
(6) Find all k e R \ {0} so that e (k − 1)z+1 dx converges. 2
(Note: this includes arguing that for the other k values the integral diverges!)
(7) Let R = {(x; y) e R2 I 
≤ y ≤ sin x & 0 ≤ x ≤ 
}. (a) Sketch R.
(b) Find the volume of the solid obtained by revolving R around the x-axis.
(c) Find the volume of the solid obtained by revolving R around the line y = 2.
(d) Find an integral which represents the volume of the solid obtained by revolving R around the y-axis.
(8) Determine if each statement is true or false. If true, give a proof. If false, explain why, or provide a counter-example.
o
(a) Suppose that f is continuous on [1; o). If lim f (x) = 0, then f (x)dx converges.
x→o 2
o
(b) Suppose that f is differentiable, and f\ is continuous. Then f\ (x)dx converges 年÷
2
f has a horizontal asymptote y = L as x - o (for some L e R).
2
(c) There exists a non-negative function f on (0; 1] such that lim f (x) = o and ef (x)dx
x→o+ o
2023-03-01