Transcript HELM Workbook 27 (Multiple Integration) EVS Questions

Evaluate without integration:
x 3 y  4
  2dydx
x 0 y  2
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6
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12
2
12
6
21
Don’t know
2
1.
2.
3.
4.
5.
Evaluate without integration:
y  2 x 11
  dydx
y 1 x  7
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14
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7
4
7
14
22
Don’t know
4
1.
2.
3.
4.
5.
1.
Which of the following integrals
does not make sense?
2 3 z
   f ( x, y, z )dxdzdy
1 y 1 0
2.
4 x2 x4
   f ( x, y, z)dydxdz
3.
0
4.
9
1 y 2 x 2
0
   f ( x, y, z)dzdxdy
3
0
0
0
0
0
4
1  1 z 2
 f ( x, y, z)dxdydz
3
 
1 z 2  y 2
2
1 z 2
2
1
1
0 1
b d
g
(
x
)
h
(
y
)
dydx
can be written as

a c
b
d
a
c
 g ( x)dx  h( y )dy.
1. True
2. False
3. Don’t know
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ls
e
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on
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k
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Tr
ue
0
What physical quantity does the
surface integral  f ( x, y)dA represent if
A
f(x, y)=1?
1. Integral represents the
mass of a plane lamina
of area A.
2. Integral represents the
moment of inertia of the
lamina A about the xaxis.
3. Integral represents the
area of A.
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3
What physical quantity does the
surface integral  f ( x, y)dA represent if
f(x, y)=y2ρ(x,y)?
A
1. Integral represents the
mass of a plane lamina
of area A.
2. Integral represents the
moment of inertia of the
lamina A about the xaxis.
3. Integral represents the
area of A.
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2
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3
What physical quantity does the
surface integral  f ( x, y)dA represent if
A
f(x, y)=ρ(x,y)?
1. Integral represents the
mass of a plane lamina
of area A.
2. Integral represents the
moment of inertia of the
lamina A about the xaxis.
3. Integral represents the
area of A.
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2
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3
If you change the order of integration,
which will remain unchanged?
1. The integrand
2. The limits
3. Don’t know
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4 3
Evaluate I   3 y  2dydx.
2 1
24
32
44
56
Don’t know
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44
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32
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24
1.
2.
3.
4.
5.
4
Evaluate I    1  2 cos  ddr.
1 0
1.
2.
3.
4.
5.
3π-12
3π
5π
3π+12
Don’t know
Evaluate  16 x
V
2
yzdV
region enclosed by
0  x  2,0  y  1,0  z  3.
3
6
9
12
None of these.
0
on
e
of
th
es
12
e.
0
N
0
9
0
6
0
3
1.
2.
3.
4.
5.
where V is the
Which diagram best represents the
1
area of integration of
1.
x
.


3
y

2
x
dydx

0 0
2.
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k
on
D
Don’t know
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2
3.
1
0%
Which diagram best represents the
1 x2
area of integration of
.


3
x

2
y
d
ydx

2
0 0
3.
0%
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4
0%
3
0%
2
4.
2.
1
1.
Which diagram best represents the
x2 y  x
region or integration of 
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4
4.
x 1 y 1
3
3.
.


y

x
dydx

2
2.
1
1.
2
Which diagram best represents the
x
region or integration of   
y

0
x
2.
1.
1 1

dydx.

4.
0
4
0
3
0
2
0
1
3.
2
Which diagram best represents the
3

region or integration of
.
(
3
x

y
)
dydx

2
1 6 2 x
4.
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4
0%
3
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2
3.
2.
1
1.
6
What double integral is obtained
when the order of integration is
x 3 y  x
  ( y  3)dydx?
reversed
1.
x 0 y 0
y 3 x y 2
y  3 x 3
  ( y  3)dxdy
y 0 x  y 2
y 3 x y 2
  ( y  3)dxdy
y 0 x 0
4.
y  3 x 3
  ( y  3)dydx0%
y 0 x  y 2
0%
0%
0%
4
2.
3.
3
x 3
2
y 0
1
  ( y  3)dxdy
What double integral is obtained
when the order of integration is
x  3 y  3 x
reversed
1.
y  3 x

 x)dydx ?
x 0 y 0
x 3
2
(
y
  x)dxdy
y 0 x 0
 (y
2
2.
y  3 x  3 y

2
(
y
  x)dxdy
y 0 x 0
3.
y 3 x 3
4.
y  3 x  3 y

2
(
y
  x)dxdy
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4
3
y 0 x  y
2
y  0 x  3 y
1

2
(
y
  x)dxdy
What double integral is obtained
when the order
of
integration
is
3 6
  (3 x  y
reversed

3
2
(
3
x

y
)dxdy

62 x 0

3
2
(
3
x

y
)dydx

0 3 y
2
4.
6

3
2
(
3
x

y
)dxdy

0 3 y
2
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6
6
3
3.
2.
2
0 3 y
2
1

6
2
(
3
x

y
)dxdy

)dydx ?
0 6 2 x
1.
3
2
Which of the following integrals
3 7 y
1.
7 3 y
are equal to    f ( x, y, z)dzdydx?
   f ( x, y, z)dzdxdy
2.
1 1 1
1 1 1
7 3 y
3.
3 7 y
   f ( x, y, z)dzdydx
1 1 1
  f ( x, y, z)dxdydz
4.
1 1 1
3 7 7
1 1 1
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5
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4
   f ( x, y, z)dydzdx
0%
3
1 1 z
2
3 7 z
   f ( x, y, z)dydzdx
1
5.
Which of the following integrals is
3 4x
0 0
4x 3
  f ( x, y)dxdy
0 3
5.
4x 3
  f ( x, y)dydx
0 0
  f ( x, y)dxdy
0 y
4
4.
y
12 4
  f ( x, y)dxdy
0%
0 0
0%
0%
0%
0%
5
y
12 4
12 3
4
3.
2.
3
0 0
2
  f ( x, y )dxdy
1
1.
equal to   f ( x, y)dydx ?
Which dose not describes the graph
of the equation r=cos θ?
Line
Circle
Spiral
Rose
os
e
0%
R
ir a
l
0%
Sp
ir c
le
0%
C
ne
0%
Li
1.
2.
3.
4.
Convert the integral to polar
2
2 a 2 ay  x
2
coordinates   x dydx :
0
0
1.
 2 a sin 

2
2
r
cos
drd

0
0
2.

2 2 a sin 
3.

3
2
r
cos
d dr

0
0

2 2 a sin 
4.
 r
0
0
3
cos drd
3
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4
2 2 a sin 
3
0
2
0
1

3
2
r
cos
drd

Convert the integral to polar
a
a2  x2
0
0
coordinates   3xdydx :
1.
 a
2
3
r
  cosdrd
2.
0 0
 a
3.
  3rdrd 
0 0
2 a
2
r
  cos drd 
4.

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4
0 0
cosdrd
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3
  3r
2
2
2 a
1
0 0
Integrate the function f ( x, y)  x  xy over
the part of the quadrant
2
2
x  0, y  0, x  y  1 in polar coordinates.
3
2
1.

2 1
4
r
  cosdrd
0 0
2.

2 1
3.
3
r
  cosdrd
0 0
 1
4
r
  cos drd 
0 0
0%
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0%
4
3
r
  cos drd 
0%
3
 1
2
4.
1
0 0
Which of the following integrals
2 
is equivalent to   rd dr ?
1.
0
  dydx
0  4 x 2
4 x 2
0
0
4 y 2
4 y 2
  dxdy
0
0
0
0
0
0
4
2
3
0
4.
2
  dxdy
2
2
  dydx
3.
2
2.
1
2
0 0
2 3
Evaluate the integral
  2x e
3  y2
dydx .
2 0
1. 0
2. 17.63218
3. Cannot be done
algebraically
0%
1
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2.
0%
3
Evaluate the volume under the surface
given by z=f(x, y)=2xsin(y) over the
region bounded above by the curve
y=x2 and below by the line y=0 for
0≤x≤1.
1.
2.
3.
4.
0.982
1.017
0.983
1.018
0%
1.
0%
2.
0%
3.
0%
4.
Evaluate f(x, y)=x2y over the
quadrilateral with vertices at (0, 0),
(3, 0), (2, 2) and (0,4)
1.
17
6
3.
113
6
2.
49
6
4.
145
6
0%
1
0%
2
0%
3
0%
4
Find the volume under the plane
z=f(x, y)=3x+y above the rectangle
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no
w
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’t
k
on
0%
D
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13
/3
0%
10
11/3
7
10
13
Don’t know
11
1.
2.
3.
4.
5.
1  y  3.
7
0  x  1,
A tetrahedron is enclosed by the planes
x=0, y=0, z=0 and x+y+z=6.
Express
this
as
a
triple
integral.
1.
6 6 x 6 x  y
   f ( x, y, z)dydzdx
0 0
0
2.
6 6 z 6 x  z
3.
   f ( x, y, z)dzdydx
0 0
6 6 6
0
   f ( x, y, z )dzdydx
0 0 0
4.
6 6 x 6 x  y
   f ( x, y, z)dzdydx
0
0
0
0%
1
0%
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2
3
0%
4
A tetrahedron is enclosed by the planes
x=0, y=0, z=0 and x+y+z=6. Find the
position of the centre of mass.
1.
3 3 3
 , , 
4 4 4
3.
3 3 3
 , , 
2 2 2
2.
1,1,1
4.
9 9 9
 , , 
4 4 4
0%
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2
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3
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4
Which of the following represents the
4 x 1
double integral
  3xydydx
after the
0 0
inner integral has been evaluated?
1.
4
3
2
3
0 2 ( x  x  x2. )dx
4
3.
4
1
2
3
(
x

x

x
)dx
0 2
3 3
0 2 ( x  x)dx4.
4
 3( x
0
3
 x)dx
0%
1
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2
0%
3
0%
4
Which of the following represents the
3 2x
double integral
  7 x ydydx after the
3
0 1
inner integral has been evaluated?
1.
3
5
3
7
(
4
x

x
)dx

0
2.
3
7
6
3
(
4
x

x
) dx
0 2
3.
3
7
5
3
0 2 (4 x  x 4.)dx
3
 7( 2 x
0
5
 x )dx
3
0%
1
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2
0%
3
0%
4
Find the moment of inertia about the
y-axis of a cube of side 2, mass M and
uniform density.
1.
8
M
3
2.
40
M
3
3.
64
M
3
4.
Don’t know
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1
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0%
2
3
0%
4
Find the centre of pressure of a
rectangle of sides 4 and 2, as shown,
immersed vertically in a fluid with one
of its edges in the surface.
1.
4

3
4.
Don’t know
0%
0%
0%
0%
4
8

1,

3


 2,

3
2.
3.
2
4

3
1
1
 ,
2
A rectangular thin plate has the
dimensions shown and a variable
density ρ, where ρ=xy. Find the centre
of gravity of the lamina.
1.
4.
4

,
2


3

5.
Don’t know
0%
0%
0%
0%
0%
5
4

3
4

1,

3

 , 2
4

3
3.
2.
2
3

4
1

1,
