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.