evaluate without integration: 1.2 2.12 3.6 4.21 5.don’t know
TRANSCRIPT
Evaluate without integration:
2 12 6 21
Don’t
know
0% 0% 0%0%0%
3
0
4
2
2x
x
y
y
dydx
1. 2
2. 12
3. 6
4. 21
5. Don’t know
Evaluate without integration:
4 7 14 22
Don’t
know
0% 0% 0%0%0%
2
1
11
7
y
y
x
x
dydx
1. 4
2. 7
3. 14
4. 22
5. Don’t know
Which of the following integrals does not make sense?
0 000
2
1
3
1 0
),,(y
z
dxdzdyzyxf
1.
4
0
2
1
4
2
),,(x x
dydxdzzyxf
2.
1
1
1
1
1
0
2
2
22
),,(z
z
yz
dxdydzzyxf
3.
9
3
1
0 0
2 2
),,(y x
dzdxdyzyxf
4.
can be written as
0 00
b
a
d
c
dydxyhxg )()(
.)()( dyyhdxxgb
a
d
c
1. True
2. False
3. Don’t know
What physical quantity does the surface integral represent if
f(x, y)=1?
1 2 3
0% 0%0%
A
dAyxf ),(
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 x-axis.
3. Integral represents the area of A.
What physical quantity does the surface integral represent if
f(x, y)=y2ρ(x,y)?
1 2 3
0% 0%0%
A
dAyxf ),(
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 x-axis.
3. Integral represents the area of A.
What physical quantity does the surface integral represent if
f(x, y)=ρ(x,y)?
1 2 3
0% 0%0%
A
dAyxf ),(
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 x-axis.
3. Integral represents the area of A.
If you change the order of integration, which will remain unchanged?
1 2 3
0% 0%0%
1. The integrand
2. The limits
3. Don’t know
Evaluate .
24 32 44 56
Don’t
know
0% 0% 0%0%0%
dydxyI 4
2
3
1
23
1. 24
2. 32
3. 44
4. 56
5. Don’t know
Evaluate .
Don’t
know
0% 0% 0%0%0%
drdI
4
1 0
cos21
1. 3π-12
2. 3π
3. 5π
4. 3π+12
5. Don’t know
Evaluate where V is the
region enclosed by .
0 0 000
V
yzdVx216
30,10,20 zyx
1. 3
2. 6
3. 9
4. 12
5. None of these.
Which diagram best represents the
area of integration of .
0% 0%0%
1.
2.
Don’t know3.
dydxxyx
1
0 0
23
Which diagram best represents the
area of integration of .
0% 0%0%0%
1. 2.
3.4.
dydxyxx
1
0 0
2
2
23
Which diagram best represents the
region or integration of .
0% 0%0%0%
1. 2.
3. 4.
dydxxyx
x
xy
y
2
1 1
2
Which diagram best represents the
region or integration of .
0 000
1. 2.
3. 4.
dydxy
x
x
1
0
1 2
Which diagram best represents the
region or integration of .
1 2 3 4
0% 0%0%0%
1. 2.
3. 4.
dydxyxx
3
1
6
26
2 )3(
What double integral is obtained when the order of integration is
reversed ?
0% 0%0%0%
dydxyx
x
xy
y
3
0 0
)3(
dxdyyy
y
yx
x
3
0 3
2
)3(
1.
dxdyyy
y
x
yx
3
0
3
2
)3(
2.dxdyy
y
y
yx
x
3
0 0
2
)3(
3.
dydxyy
y
x
yx
3
0
3
2
)3(
4.
What double integral is obtained when the order of integration is
reversed ?
0% 0%0%0%
dydxxyx
x
xy
y
3
0
3
0
2 )(
dxdyxyxy
y
x
x
3
0
3
0
2 )(
1.
dxdyxyy
y
yx
x
3
0
3
0
2 )(
2.
dxdyxyy
y
x
yx
3
0
3
3
2 )(
3.
dxdyxyy
y
yx
yx
3
0
32 )(
4.
What double integral is obtained when the order of integration is
reversed ?
0% 0%0%0%
dydxyxx
3
0
6
26
2 )3(
dxdyyxy
3
0
6
23
2 )3(
1.
dydxyxy
6
0
3
23
2 )3(
2.
dxdyyxx
6
26
3
0
2 )3(
3.
dxdyyxy
6
0
3
23
2 )3(
4.
Which of the following integrals
are equal to ?
0% 0% 0%0%0%
3
1
7
1 1
),,(y
dzdydxzyxf
7
1
3
1 1
),,(y
dzdxdyzyxf
1.
7
1
3
1 1
),,(y
dzdydxzyxf
2.
3
1
7
1 1
),,(y
dxdydzzyxf
3.
3
1
7
1
7
),,(z
dydzdxzyxf
4.
3
1
7
1 1
),,(z
dydzdxzyxf
5.
Which of the following integrals is
equal to ?
0% 0% 0%0%0%
3
0
4
0
),(x
dydxyxf
x
dxdyyxf4
0
3
0
),(
1.
12
0
3
4
),(y
dxdyyxf
2.
12
0
4
3
),(
y
dxdyyxf
3.
12
0
4
0
),(
y
dxdyyxf
4.
x
dydxyxf4
0
3
0
),(
5.
Which dose not describes the graph of the equation r=cos θ?
Lin
e
Circ
le
Spira
l
Rose
0% 0%0%0%
1. Line
2. Circle
3. Spiral
4. Rose
Convert the integral to polar
coordinates :
0% 0%0%0%
a xay
dydxx2
0
2
0
2
2
0
sin2
0
22 cosa
drdr
1.
drdra
2
0
sin2
0
23 cos
2.
2
0
sin2
0
23 cos
a
drdr
3.
2
0
sin2
0
33 cosa
drdr
4.
Convert the integral to polar
coordinates :
0% 0%0%0%
a xa
xdydx0 0
22
3
0 0
2 cos3a
drdr
1.
0 0
3a
rdrd
2.
2
0 0
2 cosa
drdr
3.
2
0 0
2 cos3
a
drdr
4.
Integrate the function over the part of the quadrant
in polar coordinates.
0% 0%0%0%
23),( xyxyxf
1,0,0 22 yxyx
2
0
1
0
4 cos
drdr
1.
2
0
1
0
3 cos
drdr
2.
0
1
0
4 cos drdr
3.
0
1
0
3 cos drdr
4.
Which of the following integrals
is equivalent to ?
0 000
2
0 0
drrd
2
0
0
4 2x
dydx
1.
2
0
4
0
2x
dydx
2.
2
2
4
0
2y
dxdy
3.
2
0
4
0
2y
dxdy
4.
Evaluate the integral .
1 2. 3
0% 0%0%
2
2
3
0
3 2
2 dydxex y
1. 0
2. 17.63218
3. Cannot be done algebraically
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% 0%0%0%
1. 0.982
2. 1.017
3. 0.983
4. 1.018
Evaluate f(x, y)=x2y over the quadrilateral with vertices at (0, 0),
(3, 0), (2, 2) and (0,4)
1 2 3 4
0% 0%0%0%
6
171.
6
492.
6
1133.
6
1454.
Find the volume under the plane z=f(x, y)=3x+y above the rectangle
11/
3 7 10 13
Don’t
know
0% 0% 0%0%0%
.31,10 yx
1. 11/3
2. 7
3. 10
4. 13
5. Don’t know
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 2 3 4
0% 0%0%0%
6
0
6
0
6
0
),,(x yx
dydzdxzyxf
1.
6
0
6
0
6
0
),,(z zx
dzdydxzyxf
2.
6
0
6
0
6
0
),,( dzdydxzyxf
3.
6
0
6
0
6
0
),,(x yx
dzdydxzyxf
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 2 3 4
0% 0%0%0%
4
3,4
3,4
31.
1,1,12.
2
3,2
3,2
33.
4
9,4
9,4
94.
Which of the following represents the
double integral after the
inner integral has been evaluated?
1 2 3 4
0% 0%0%0%
4
0
1
0
3x
xydydx
dxxxx )(2
3 324
0
1.
dxxxx )(2
1 324
0
2.
dxxx )(2
3 34
0
3.
dxxx )(3 34
0
4.
Which of the following represents the
double integral after the
inner integral has been evaluated?
1 2 3 4
0% 0%0%0%
3
0
2
1
37x
ydydxx
dxxx )4(7 353
0
1.
dxxx )4(2
7 363
0
2.
dxxx )4(2
7 353
0
3.
dxxx )2(7 353
0
4.
Find the moment of inertia about they-axis of a cube of side 2, mass M and
uniform density.
1 2 3 4
0% 0%0%0%
M3
81.
M3
402.
M3
643.
Don’t know4.
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.
0% 0%0%0%
3
4,2
11.
3
8,1
2.
3
4,2
3.
Don’t know4.
A rectangular thin plate has the dimensions shown and a variable
density ρ, where ρ=xy. Find the centre of gravity of the lamina.
0% 0% 0%0%0%
4
3,1
1.
2,4
32.
3
4,1
3.
2,3
44.
Don’t know5.