MATH 1700 ICM Summer 2013 Quiz 2A Solutions
[4] 1.
3
ba
=
n
n
x =
xi = xi = a + ix = 1 +
3i
.
n
Hence the definite integral is equal to
lim
n
n
X
n
f (xi )x
i=1
3X
= lim
n n
i=1
3i
1+
n
+
3i
1+
n
2
.
[4] 2.
x =
1
.
n
xi = xi = a + ix = a +
i
.
n
Thus a =
MATH 1700 ICM Summer 2013 Quiz 1A Solutions
[5] 1. This is of 1 form which is indeterminant. Let y = (1 + 2x)1/ tan x ,
ln(1 + 2x)
. Since this is 0/0 form, LHospitals Rule applies and we get
then ln y =
tan x
ln(1 + 2x)
lim+ ln y = lim+
x0
x0
tan x
= lim
MATH 1700 ICM Summer 2013 Quiz 4A Solutions
1
x4
[4] 1. Using integration by parts u = ln x and dv = x dx. Then du = dx and v = .
x
4
3
Z 4
x 1
x4 ln x
I=
dx
4
4 x
Z 3
x
x4 ln x
dx
=
4
4
x4 ln x x4
=
+C
4
16
[8] 2. Let x = 2 tan dx = 2 sec2 d. Hence
Z
1
d
MATH 1700 ICM Summer 2013 Quiz 3B Solutions
[7] 1. The sketch is
The points of intersection are
4y y 2 = y y 2 3y = 0 y(y 3) = 0 y = 0, 3.
Using horizontal rectangles, we can use the shell method.
The radius is 7 y and the height is 4y y 2 (y) = 3y y 2 .
MATH 1700 ICM Summer 2013 Quiz 4B Solutions
[4] 1. Using integration by parts u = x and dv = sec x tan xdx. Then du = dx and v = sec x.
Z
I = x sec x
sec xdx
= x sec x ln sec x + tan x + C
[8] 2. Let x = 2 sec dx = 2 cos d. Hence
Z
1
dx
(4 x2 )3/2
Z
2
MATH 1700 ICM Quiz 2A Solutions
[8] 1. Find the point(s) on the following parametric curve where the tangent line is vertical.
x = t3 3t2 ,
y = et t.
dy
dy
et 1
= dt = 2
.
dx
dx
3t 6t
dt
The tangent line has infinite slope if this expression goes to infin
MATH 1700 In Class Problem Workshop 6
1. Find the volume of the region R rotated about the given line.
(a) R is the region bounded by y = sin(x2 ),
y = 0 about the yaxis. (Just one part)
(b) R is the region bounded by y = x2 ,
y = 0,
x = 1,
x = 2 about x
INTERNATIONAL COLLEGE OF MANITOBA
DATE: July 13, 2013
MIDTERM 2
PAGE: 1 of 4
DEPARTMENT & COURSE NO: MATH1700
TIME: 1 hour
EXAMINATION: Calculus 2
EXAMINER: N. Harland
1. (8 points) Find the volume of the solid with a base of a circle with radius 3, cente
MATH 1700 ICM Quiz 4B Solutions
(20 minutes)
[5] 1. Set up, but do not evaluate, a definite integral or definite integrals to determine the
volume of the region bounded by y = x2 4 and x = y 2 rotated about x = 7.
A sketch is given below
The points of int
MATH1700 Key Laboratory Quiz #5B
WINTER 2015
Time: 20 minutes
Name: _
Circle your instructors name:
Student ID: _
Guye Harland
Value: 22 marks
1. For each of the following improper integrals, find its value if it exists
a)
1
9 + 4
Solution:
1
1
1
=
+
MATH 1700 ICM Summer 2013 Quiz 2B Solutions
[4] 1.
ba
5
=
n
n
x =
xi = xi = a + ix = 2 +
5i
.
n
Hence the definite integral is equal to
n
X
lim
n
n
f (xi )x
i=1
3X
= lim
n n
i=1
5i
2+
n
2
5i
4 2+
.
n
[4] 2.
x =
1
.
n
xi = xi = a + ix = a +
i
.
n
Thus a =
MATH 1700 Problem Workshop 8
Evaluate the following integrals
Z
1
1.
dx
x 4 x2
Z
9 x2
2.
dx
x4
Z 2
x 4
3.
dx
x3
Z
1
dx
4.
2
x 6x + 13
Z
x5
dx
5.
x2 + 9
Z /2
cos t
dt
6.
1 + sin2 t
0
Z
5x2 + 3x 2
7.
dx
x3 + 2x2
Z 3
x 4x 10
8.
dx
x2 x 6
Z
1
9.
dx
4
x 1
Z 1
MATH 1700 ICM Summer 2013 Quiz 5B Solutions
[7] 1. Evaluate the following integral if it converges:
Z
4
x3 ex dx
1
t
Z
t
4
x3 ex dx
I = lim
1
Z
t4
1
ew dw using w = x4
4
1
t4
1
= lim ew
t
4 1
1 t4 1 1
= lim e + e

t
4
4
1
= e1
4
= lim
t
Hence the int
MATH 1700 ICM Quiz 4A Solutions
(20 minutes)
[5] 1. Set up, but do not evaluate, a definite integral or definite integrals to determine the
volume of the region bounded by x = y 2 + 4 and y = x 6 rotated about y = 5.
A sketch is given below
The points of
MATH 1700 ICM Summer 2013 Quiz 1B Solutions
[5] 1. This is of 1 form which is indeterminant. Let y = (3x + 1)cot x ,
ln(3x + 1)
. Since this is 0/0 form, LHospitals Rule
then ln y = cot x ln(3x + 1) =
tan x
applies and we get
lim+ ln y = lim+
x0
x0
= lim+
MATH 1700 ICM Summer 2013 Quiz 3A Solutions
[7] 1. The sketch is
The points of intersection are
y 2 3 = 2y y 2 2y 3 = 0 (y 3)(y + 1) = 0 y = 1, 3.
Using horizontal rectangles, we can use the washer method.
The outer radius is 7 (y 2 3) and the inner radiu
MATH 1700 Problem Workshop 9
1. Evaluate the following integrals
Z
(a)
x sec x tan xdx
Z
(b)
x2 tan1 xdx
Z
3
x+1
(c)
dx
3
x1
Z
x3
dx
(d)
(x2 + 1)2
Z
1
(e)
dx
x4 + x2
Z 2
x 4
dx
(f)
x2
Z
1
(g)
dx
x + x2 + 4
2. For the following indefinite integrals, deter
MATH 1700 ICM Summer 2013 Quiz 5A Solutions
[7] 1. Evaluate the following integral if it converges:
Z
1
t
Z
I = lim
t
1
Z
t
e x
dx
x
t
= lim
e x
dx
x
2ew dw using w = x
1
t
= lim 2ew 
1
t
t
1
= lim 2e
+ 2e
t
= 2e1
Hence the integral converges to
2
.
MATH 1700 In Class Problem Workshop 10
1. For the following indefinite integrals, determine whether or not it converges and if it
converges, find its value.
Z 2
1
dx
(a)
2x x2
0
Z 1
ln xdx
(b)
0
Z
/2
cos x
dx
(1 sin x)2/3
0
Z e
1
dx
(d)
1 x ln x
Z 1
1
(e)
INTERNATIONAL COLLEGE OF MANITOBA
DATE: June 15, 2013
MIDTERM
PAGE: 1 of 3
DEPARTMENT & COURSE NO: MATH1700
TIME: 1 hour
EXAMINATION: Calculus 2
EXAMINER: N. Harland
[6] 1. Find the following limit.
lim+ x
x
x0
Let y = x
x
. We first find
lim+ ln y = lim+
MATH 1700 ICM Quiz 2B Solutions
[8] 1. Find the point(s) on the following parametric curve where the tangent line is horizontal.
x = e2t 2t,
y = t3 6t2 .
dy
dy
3t2 12t
= dt =
.
dx
dx
2e2t 2
dt
The tangent line is horizontal slope if this expression is zer
MATH1700 Laboratory Quiz #3B
WINTER 2015
Time: 15 minutes
Name: _
Circle your instructors name:
Student ID: _
Gueye Harland
Value: 10 marks
1. Find the area of the region R bounded by = 6 2 and =
Lets find the intersection point
= ( )
+ =
( + )( ) = (
MATH1700 Key Laboratory Quiz #5A
WINTER 2015
Time: 20 minutes
Name: _
Circle your instructors name:
Student ID: _
Guye Harland
Value: 22 marks
1. For each of the following improper integrals, find its value if it exists
2
a)
Solution:
=
+ ( . )
MATH 1700 Problem Workshop 8 Solutions
1. Using x = 2 sin dx = 2 cos d.
Z
1
dx
2
Z x 4x
2 cos d
p
=
2 sin 4 (2 sin )2
Z
2 cos d
p
=
2 sin 4 4 sin2
Z
2 cos d
=
2
sin
4 cos2
Z
2 cos d
=
4 sin cos
Z
1
=
csc d
2
1
= ln csc + cot  + C
2
I=
Using sin =
x
2
MATH 1700 ICM Winter 2014 Quiz 1B Solutions
sin1 x
x0 ln(1 x)
This is a 0/0 form, and therefore we can use LHoptials Rule to get
[6] 1. Find lim
+
sin1 x
lim
=H lim
x0+ ln(1 x)
x0+
=
1
1x2
1
1x
1
102
1
1(0)
1
1
= 1
=
[8] 2. Find lim (2x + 3)1/x
x
Let y
INTERNATIONAL COLLEGE OF MANITOBA
DATE: November 8, 2013
TERM TEST 2
PAGE: 1 of 5
DEPARTMENT & COURSE NO: MATH1700
TIME: 1 hour
EXAMINATION: Calculus 2
EXAMINER: N. Harland and Y. Maddahi
1. Evaluate the following integrals.
e
ln x
dx
[6]
(a)
1 2 x
Using
INTERNATIONAL COLLEGE OF MANITOBA
DATE: October 11, 2013
MIDTERM
PAGE: 1 of 5
DEPARTMENT & COURSE NO: MATH1700
TIME: 1 hour
EXAMINATION: Calculus 2
EXAMINER: N. Harland and Y. Maddahi
[6] 1. Evaluate the following limit.
lim (ln x)1/(xe)
+
xe
1
xe
Let y =
MATH 1700 Final Practice Questions.
1. Find limx 1 + 3 sin
2
x
x
2. The curve C is decribed in parametric equations by x(t) = t+2 cos t,
where 0 t .
y(t) = t2 sin t
(a) Find the equation of the tangent line to C where t = 0.
(b) Find the point(s) on C whe