Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
Test 1 solution:
1. a) Find the center and radius of the sphere 4 x 2
4 y2
4 z 2 8 x 16 y 1 0 .
Show work here:
4x2
x2
4 y2
y2
4 z 2 8 x 16 y 1 0
z 2 2 x 4 y 1/ 4 0
( x 1)2 ( y 2) 2
z2
21
2
b) Find the center and radius of the sphere 4 x 2
21/ 4
The center is at (1, -2, 0), and radius
4 x2
4 y2
x2
y2
4 y2
4 z 2 24 x 16 y 8 z 44 0
z 2 6 x 4 y 2 z 11 0
( x 3) 2 ( y 2) 2 ( z 1) 2
The center is at (3, -2, 1), and radius 5
2. a) Given a vector r
the given vector.
Show work here: u
4 z 2 24 x 16 y 8 z 44 0 .
25
4, 4,3 . Find a vector of magnitude 10 in the direction of
10
41
4, 4,3
b) Find x and y components of a vector of magnitude 8 and an angle of
3
with
the positive x direction.
Show work here: x 8cos
3
4, y 8sin
4
4 3
3. a) Test if the following vectors are coplanar: a
1, 4, 7 , b
c
0, 9,18 . Use the concept of scalar triple product.
Show work here:
1 4
7
a (b c )
2
1
4
0
9 18
2, 1, 4
and
0, yes coplanar
b) Test if the following vectors are coplanar: u i 4 j 7k , v
w 9 j 18k . Use the concept of scalar triple product.
2i
j
4k and
Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
Show work here:
u (v w)
1
2
4
1
7
4
0
9 18
0, yes coplanar
5, 6, 6 onto Q 1, 4,8 . Remember
4. Find orthogonal projection of P
that the orthogonal projection of a vector b onto a vector a is defined and
denoted by ortha b b proja b
Show work here:
ProjQ P
Orth Q P
P Q
Q
2
Q
29
1,
81
P ProjQ P
4, 8
376 / 81,
602 / 81, 254 / 81
5. a) Find the area of the parallelogram with vertices P(0, 0, 0), Q(-5, 1, 3),
R(-5, 0, 1) and S(-10, 1, 4).
Show work here: PQ
5, 1, 3 , PR
i
j
PQ PR
5, 0, 1
k
5
1
3
5
0 1
i 10 j 5k
Area=|PQ PR|= 126 sq. unit
b) Find the area of the triangle with vertices P(5, 4, -5), Q(3, 1, -6), and R(2, 2, -8).
Show work here: PQ
2, 3, 1 , PR
i
j k
PQ PR
2
3
3
2
3,
1
3
2, 3
7i 3j-5k
1
Area= |PQ PR|= 83 / 2 sq. unit
2
6. For what values of x are the vectors a
5, 1, x and b
Show work here: For orthogonality a b = 25+1+x 2
Therefore is no such x value we have.
5, 1, x orthogonal?
0 has no real solution.
Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
7. a) Find the distance of the point P( 5, 3,1) from the line which passes through the
two given points Q(0, 2,1) and R( 3,0, 1) .
Show work here:
i
PQ QR
Distance d =
j
k
5 1
3 2
0
2
2i+10j+13k
|PQ QR|
= 273/17 unit
QR
b) A man started walking from a point (-3, 1) and reached at the point (-2, 5). The man
started walking in the direction of the vector a 5, 2 and changes his direction
only once, when he turns at a right angle. What are the coordinates of the point
where he makes the turn? Round your answer to two decimal places.
Show work here: r (t )
3 5t,1 2t and the distance d is given as
2
2
D d
(1 5t ) (4 2t ) 2 has minimum at t 0.4483 (check it)
The required point where he turns at right angle is ( -0.76, 1.90)
8. A horizontal clothesline is tied between 2 poles, 18 meters apart. When a mass of
2 kilograms is tied to the middle of the clothesline, it sags a distance of 2 meters.
What is the magnitude of the tension on the ends of the clothesline?
T
Solution:
| T | cos i | T | sin j =|T|( 9 / 85i+2/ 85j)
W=2 9.8=2T
2
85
T
45.18
T
T
2 kg
9. Find the parametric equations of the line through the point (0,1, 2) that is
perpendicular to the line x 1 t, y 1 t, z 2t and parallel to the plane
x y z 2 0.
Show work here: u
1,1,1
The equation of the line is
1, 1, 2
L
0,1, 2
3, 1, 2
t
x 3t , y 1 t , z
3, 1, 2
2 2t
Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
10. Consider the path r (t ) (18t , 9t 2 , 9 ln t ), t 0 . Find the length of the curve
between the points P(72,144,9ln 4) and Q(18,9,0)
Solution:
r (t )
18,18t ,9 / t , | r (t ) | 9(2t 1/ t )
4
L
9(2t 1/ t )dt
9(15 ln 4) 147.48
1
Test 2 solution
xe xy .
1. A) Let f ( x, y )
a) Find
f (1,1) and b) Du f (1,1) in the direction of v
Show work here:
f ( x, y )
e xy (1 xy ), x 2e xy , f (1,1)
Du f
a) Find
Answer: a)
u
v
|v |
3 / 2, 1/ 2
3e 1/ 2e
xe xy .
B) Let f ( x, y )
u
2e, e ,
j , where u
i 3
f (1,1) and b) Du f (1,1) in the direction of v
i
j 3 , where
v
|v |
f (1,1)
b) Du f (1,1) e( 3 / 2 1)
2e, e
2
2
2. A) Find the equation of the tangent plane to z f ( x, y ) e x y at the point
(1, 1,1) .
And approximate the value of f (1.1, 0.98) using linear approximation.
Show work here:
f ( x, y) 2x 2 y 1, f (1.1, 0.98) 2(1.1) 2( 0.987) 1 1.24
2
2
B) Find the equation of the tangent plane to z f ( x, y ) e x y at the point
(1, 1,1) . And approximate the value of f (0.97, 1.2) using linear approximation.
Show work here:
fx
2 xe x
2 ye x
fy
z
2
f
y2
2
f x (1, 1)
y
2
2
f y (1, 1)
1 2( x 1) 2( y 1)
f (0.97, 1.2)
2
2x 2 y 1
2(0.97) 2( 1.2) 1 0.54
Mat 267 Engineering Calculus III Updated on 04/30/2011
3. a) Suppose that z
er cos , r
Dr. Firoz
s 2 t 2 . Find
st ,
z
z
(1,1) ,
(1,1) round
t
s
your answer to two decimal places.
Solution:
z
( s, t )
t
se r cos
z
(1,1)
t
1.47
z
(1,1) ter cos
s
e r ( sin )
t
s2 t 2
s
er ( sin )
s
2
t2
1.47
b) Let W (s, t ) F (u(s, t ), v(s, t )), where W , u and v are differentiable,
u (1, 0) 2, us (1, 0)
2, ut (1, 0) 6, v(1, 0) 3, vs (1, 0) 5, vt (1, 0) 4 . Further
1, Fv (2,3) 10 . Find Ws (1, 0) , Wt (1, 0) .
given that Fu (2,3)
Solution:
W
W u
(1,0)
s
u s
W
Wt (1, 0)
t
4. A) Given f ( x, y )
Show work here:
dz
1( 2) 10(5) 52
1(6) 10(4) 34
y 2 . Find the differential dz in terms of x and y.
2xdx 3 ydx 3xdy 2 ydy (2x 3y)dx (3x 2 y)dy
B) Given f ( x, y )
dz
x 2 3xy
W v
v s
x 2 3xy
y 2 . Find the differential dz in terms of x and y.
2xdx 3 ydx 3xdy 2 ydy (2x 3 y)dx ( 3x 2 y)dy
5. The dimensions of a closed rectangular box are measured as 90 cm, 50 cm,
and 60 cm respectively with the error in each measurement at most 0.2 cm.
Use differential to estimate the maximum error in calculating the surface area
of the box. Do not forget the unit.
Solution:
S 2lw 2lh 2wh
dS 2ldw 2wdl 2ldh 2hdl 2wdh 2hdw
2(90 50 90 60 50 60)0.2 160 sq.cm
Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
6. Verify if the function u( x, t ) sin( x at ) satisfies the wave equation
2
u
2
t
Show work here:
2
a2
u
x2
0.
u
t
u
x
a cos( x at ),
2
u
t
2
a 2 sin( x at )
2
u
x2
cos( x at ),
2
t
u
2
sin( x at )
2
a
2
u
x2
0 , yes, verifies.
7. Find all the relative maximum and minimum and saddle points on the given
surface z f ( x, y ) x 4 y 4 4 xy 1 . Do not forget to check the sign of the
discriminant at each stationary point. Write your answer in the box below.
Answer:
fx
4 x3 4 y
f xx
12 x 2 , f yy
0, f y
4 y3 4x
12 y 2 , f xy
0,
4, D
f xx f yy [ f xy ]2
We have the solution (0, 0), (1, 1) and (-1, -1) as stationary points
Check that saddle point at (0, 0, 1), min at (1, 1) and (-1, -1) and no max.
( x 2 y )dA , where D is the region bounded by the
8. Evaluate the integral
D
parabolas y 2 x 2 , y 1 x 2 . You may round your answer to two decimal
places. Put your answer in the box below
Solution:
1 1 x2
( x 2 y )dA
D
9.
x 2 ydydx
1 2x
2.13
2
A) Evaluate the integral by reversing the order of the integral
Write your answer in the box below.
Solution:
3 x/3
3
1 3
2
2
x x2
e x dxdy
e x dydx
e dx 1350.35
0 3y
3
0 0
0
1
3
0 3y
2
e x dxdy .
Mat 267 Engineering Calculus III Updated on 04/30/2011
B)
8
0
2 x3
2
3
2
x4
y
4
4
e x dydx
e dxdy
x 3e x dx
0 0
10. A) Find
2221527.38
0
1
1 x2
1
Dr. Firoz
1 x2
1 0
dydx using polar coordinates. Write answer
y2
in the box below.
x2
Solution:
1
1 x
1 0
1 x
2
y
r 2 drd
dydx
2
0
128 / 9 14.22
0
( x 2) 2
B)
4
0
4x x
1
2
0
x2
1
/ 2 4cos
1
2
y2
1
y 2 dydx
1 r2
0 0
rdrd
y2
4
1.3
Test3 solution:
Write down the double integral and also triple integral to find the volume of the
tetrahedron bounded by the coordinate planes and the plane z 4 4x 2 y .
Evaluate both the integrals.
2
2
D
E
y 2 2x
x
y
x
1 2 2x
zdA
Complete the limit:
D
zdxdy
D
(4 4 x 2 y )dydx
0
4 / 3 1.33
0
1 2 2x 4 4x 2 y
dV
Complete the limit:
E
dzdydx
E
dzdydx
0
0
4 / 3 1.33
0
2. Find the volume of the solid that lies within the sphere x 2
the xy-plane, and below the cone z
x2
y2
y2
z2
4 , above
Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
Solution: Draw the picture and identify the region.
/2 2
V
2
2
dV
E
8
sin d d d
2
11.85
3
/4 0 0
3. A sphere has center at (0, 0, 0) and radius equal to m. Use spherical coordinate to
evaluate the volume of the sphere
Solution: the sphere has the equation x 2
2
y2
z2
m2
m
2
V
4 m3
3
sin d d d
0 0 0
4.19m3
sin xdx cos ydy , where C is the arc consisting of the
4. Evaluate the line integral
C
top half of the circle x 2 y 2 1 from (1, 0) to (-1, 0) and the line segment from
(-1, 0) to (-2, 3). Write exact answer, no decimals.
Solution: Check that on C1 curve: x cos t, y sin t, 0 t
on C2 curve: x
1 t, y 3t, 0 t 1
sin xdx cos ydy
C
sin xdx cos ydy
C1
C2
and
C1
sin xdx cos ydy
C2
1
sin(cos t )( sin t )dt cos(sin t )(cos t )dt
0
sin( 1 t )( dt ) cos(3t )3dt
0
cos1 cos 2 sin 3
5. a) Show that the line integral (1 ye x )dx e x dy is independent of path, where
C
C is any path from (0, 1) to (1, 2).
b) Use line integral with parametric representation for C to evaluate integral.
Solution: P 1 ye x , Q e
x
Q
x
P
y
e x , the path C is independent.
1
(1 ye x )dx e x dy
C
(1 (1 t )e
t
e t )dt
2/e
0
Where C has the parametric representation: r (t )
0,1
t (1,1)
x t, y 1 t
Mat 267 Engineering Calculus III Updated on 04/30/2011
6. Use Greens’ theorem to evaluate the line integral
Dr. Firoz
x 4 dx xydy , where C is the
C
triangular path consisting of the line segments from (0, 0) to (1, 0), from (1, 0) to
(0, 1), and from (0, 1) to (0, 0).
(0, 1)
11 x
Q
P
dA
ydydx 1/ 6
Solution: x 4 dx xydy
x
y
C
D
0 0
y 1 x
Q P
Where P x 4 , Q xy
(0,0)
(1,0)
y
x
y
7. Show that the vector field F ( x, y, z ) y 2 i (2 xy e3 z )j 3 ye3 z k is conservative.
Also find the scalar function f ( x, y, z) such that grad f F .
Solution:
i
Curl F
j
k
o the field is conservative
F
x
y
z
2
3z
y 2 xy e
3 ye3 z
For scalar function we use the following results;
f x y 2 f y 2 xy e3 z , f z 3 ye3 z
f
xy 2 h( y, z )
Now f
xy 2
fy
ye3 z
2 xy hy ( y, z )
g ( z)
fz
3 ye3 z
2 xy e3 z ,
g ( z)
xy 2
Thus the scalar function is f ( x, y, z )
ye3 z
8. Find the flux of the vector field F ( x, y, z)
center at the origin.
h( y , z )
S
g (z)
g ( z) k
C
zi yj xk across the unit sphere
Solution: Use x sin cos , y sin sin , z cos ,0
r ( , ) xi yj zk i sin cos
j sin sin
k cos ,
2
2
r r i sin cos
j sin sin
k sin cos
F (r ( , ) i cos
j sin sin
Now
F ( x, y, z ) dS
ye3 z
,0
2
and
k sin cos
F n dS
D
2
F (r
D
4 / 3 4.19
Check the integral.
(2sin 2 cos cos
r )dA
0 0
sin 3 sin 2 ) d d
Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
Quiz 1: February 11, 2011
1. Name
2. Find the equation of the largest sphere center at (2, 4, 6) contained in the first
octant.
Solution: The sphere contained in the first octant must have radius 2 units. The
equation of the sphere is ( x 2) 2 ( y 4) 2 ( z 6) 2 4
3. A man walks due east on a deck of a ship at 5 miles per hour. The ship is moving
north at a speed of 11 miles per hour. Find the speed and direction (in radians) of
the man relative to the surface of the water.
Solution: u
5,0 , v
0,11
The resultant velocity is w u v
52 112
The speed | w |
The direction with north:
5,0
0,11
5,11
(0,11)
146
cos
112
1
146 112
(5,0)
0.43
Quiz 2: March 25, 2011
Set A
2 3
1. Evaluate
0
xy 2
dydx
1 x2
3
2
3
x
dx y 2 dy
2
1 x
0
3
9 ln 5
x 2 3 y 2 dA , where D {( x, y) | x 0, y 1, y
2. Evaluate
x, z 0}
D
1 y
x
2
2
( x 2 3 y 2 )dxdy 5 / 6
3 y dA
D
y=x
0 0
1 1
3
e x dxdy by changing the order of integration.
3. Evaluate
0
y
1 x2
1 1
x3
3
e x dydx
e dxdy
0
y
0 0
e 1
3
3
3 x x2
x2
4. Evaluate by using polar coordinate:
0
problem number 10(B) in test 2.
0
y 2 dydx
6 , compare with
Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
Set B
2 3
3
x2 y
dxdy
1 y2
3
1. Evaluate
0
2
x 2 dx
3
y
dy
1 y2
0
9 ln 5
1 1
3
e y dydx by changing the order of integration.
2. Evaluate
0
1 1
3
e y dydx
0
x
x
e 1
3
compare with set A.
x 2 3 y 2 dA , where D {( x, y) | x 0, y 1, y
3. Evaluate
x, z 0}
D
1 y
x
2
2
( x 2 3 y 2 )dxdy 5 / 6
3 y dA
D
0 0
2
2 x x2
x2
4. Evaluate by using polar coordinate:
0
2
2x x
2
0
/ 2 2cos
x2
0
y 2 dydx
y 2 dydx
0
r 2 drd
0
16 / 9
0
Quiz 3: April 22, 2011
1. Given that r ( x, y, z) xi yj zk , use spherical coordinate for a unit sphere
center at origin to find the normal vector r r .
Solution:
r ( x, y , z )
xi yj zk
r ( , ) i sin cos
j sin sin
r ( , ) i cos cos
j cos sin
r( , )
r
r
i sin sin
i sin 2 cos
k cos
k sin
j sin cos
j sin 2 sin
k sin cos
2 3/ 2
(x
y 3/ 2 ), 0
3
S
Round your answer correct to three decimal places.
ydS , where S is the surface z
2. Evaluate
Solution:
1 1
ydS
S
y 1 y x dA
D
y 1 x
0 0
ydxdy
0.733
x 1, 0
y 1.
Mat 267 Engineering Calculus III Updated on 04/30/2011
3. Find the flux of the vector field F( x, y, z)
center at the origin.
Dr. Firoz
zi yj xk across the unit sphere
Use x sin cos , y sin sin , z cos ,0
,0
r ( , ) xi yj zk i sin cos
j sin sin
k cos ,
2
2
r r i sin cos
j sin sin
k sin cos
F (r ( , ) i cos
j sin sin
Now
F ( x, y, z ) dS
S
2
and
k sin cos
F n dS
D
2
F (r
(2sin 2 cos cos
r )dA
D
sin 3 sin 2 ) d d
0 0
4 /3
Check the integral.
Pop Quiz
Answers to selected homework problems:
Section: 12.5
10.
4
3
14.
2
15
16.
27
8
20. 128
18. 144
32.
Look at the domains:
y
z
z
x
x
A
B
From diagram A: Domain D is xy plane
1 1 x 1 x2
0 0
1 1 y 1 x2
f ( x, y, z )dz ]dydx
0
[
f ( x, y, z )dz ]dxdy
0 0
0
From diagram B: Domain D is xz plane
1 1 x2 1 x
[
0
0
1 1 z 1 x
f ( x, y, z )dy ]dzdx
0
2y
y2
z 1 x2
y 1 x
[
z
[
0
0
f ( x, y, z )dy ]dxdz
0
From diagram C: It has two separate regions for domain D
y
C
Mat 267 Engineering Calculus III Updated on 04/30/2011
11
1 z
1 z
[
0
0
1
1
f ( x, y, z )dx]dydz
0
01
[
0
1 y
[
2
1 2y y 1 y
0
Dr. Firoz
1 z
1
f ( x, y, z )dx]dydz
0
1
f ( x, y, z )dx]dzdy
1 z
[
0 2 y y2
0
f ( x, y, z )dx]dzdy
0
Section: 12.6
8. A hyperboloid of one sheet with axis the z axis.
12. A cone opening upward in the first octant. 16. 7
20.
65
4
22.
4
(8 33/ 2 )
3
28.
18.
2
35
162
5
Section: 12.7
8. the surface of a sphere of radius 1 center at (0, 0, 1)
10. a)
26.
2 b)
8 2
3
24.
sec2 cot csc
2
(5e3 2)
36. 0
Chapter 13
Section: 13.1
4. Choose different x and y values and the length of the vector, then plot them (look at
example 1 in your text book)
y
x
y xy
y
1
1
24. cos
26. f
sin
sin
( x y)
( x y ) , plot yourself
2
2
z
z z
z
2
2
Section: 13.2
6. cos1 cos 2 cos3
10.
1 3/ 2
(14
1)
6
16. On C2 negative, on C1 positive 20.
34.
Section: 13.3
2. 6
8. f ( x, y ) x 2 y x ln x K
16. a) f ( x, y, z ) e y x ze z
b) 2e
14. 35/3
18. 2/e
1
(15 cos1 cos 4)
2
Mat 267 Engineering Calculus III Updated on 04/30/2011
Dr. Firoz
Section: 13.4
2.
2
10. 0
14. -16
18. 12
16.
Section: 13.5
4. a) x(sin xz cos xy ) y cos xy
8. a) greater than zero b) is zero
16. f ( x, y, z) sin xy cos z K
z sin xz
b) 0
10. Try yourself
Section: 13.6
x2 y 2
1,0 z 2 16. x
4 9
s sin , z 3 s cos , 0 s 1; 0
1
38.
(263/ 2 103/ 2 )
24
2.Portion of elliptic cylinder
22. x
s cos , y
30. x y 2z
0
x, y
y, z
1 2x2 4 y 2
24. 0
34 108
2
Section: 13.7
6.
4
3
8.
6
6
14.
4
(8
2)
22.
3
Concept:
1. Evaluation of double integrals and triple integrals
2. Cylindrical and spherical coordinates
3. Vector fields, scalar functions, line integral and fundamental theorem of line
integral, conservative vector field,
4. Greens’ theorem
5. Curl and divergence
6. Parametric surfaces and surface integrals
2.