Calculus 2 Chapter13(6)

5/14/2015

1

MULTIPLE INTEGRALS

13.1: DOUBLE INTEGRALS

Let be a function of two variables that is defined on region

The double integral of over is defined by

Where

Theorem: (Fubini’s Theorem)

Suppose that is integrable over the rectangle

Then we can write the double integral of over as either of

the iterated integrals:

f .Rf R

or .dA dxdy dA dydx

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,R x y a x b and c y d

Rf

f

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MULTIPLE INTEGRALS


Def: Let be a continuous function of two variable such that

for every . Then the volume of the solid

that is lies over a region is

Properties of double integrals:

,R

V f x y dA

f

,x y R


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R

1. , ,

2. , , , ,

3.If ( , ) 0, then , 0

R R

R R R

R

c f x y dA c f x y dA

f x y g x y dA f x y dA g x y dA

f x y f x y dA

, 0f x y

MULTIPLE INTEGRALS


Ex:

Sol:


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1 2

1 24. If , then , , ,R R R

R R R f x y dA f x y dA f x y dA

4 2 4 2

2 3 2 3

1 0 1 0

2 24 43 2 33 2 3

1 1 00

, 6 4 6 4

6 42 2

3 2

R

f x y dA x xy dxdy x xy dx dy

x x ydy x x y dy

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MULTIPLE INTEGRALS


We leave it as an exercise to show that you get the same

value by integrating first with respect to y, that is, that

4

3 2 3

1

2 2 2 2 0y dy


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MULTIPLE INTEGRALS


EX: Evaluate

Ex: Let R be the region bounded by the graphs of y = x, y = 0

and x = 4. Evaluate

Changing the Order of the double integrals:

To change the order of integration in a multiple integral, follow

these steps:

Step 1: Sketch the region of integration. Prepared by Dr. F.G.A

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1 1

2

0

24x

x ydydx

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MULTIPLE INTEGRALS


Step 2: Slice up the region according to the new order, and

determine the new limits of integration one at a time, starting

with the outer variable.

Step 3: Evaluate the new integral.

EX: Change the order of the integration Sol:

It is clear that

Looking at the graph, the lowest value of y

is 0 and the highest value of y is 2. Thus,

For x, the highest value is 1, and since

, the limits of y is

Hence, we get Prepared by Dr. F.G.A

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1 2

0 0

,

x

f x y dydx

0 1 and0 2 .x y x

0 1

2y x

0 2.y

1 12

y x 12

x y

MULTIPLE INTEGRALS


EX: Change the order of the integration

Sol:

It is clear that

Looking at the graph, the lowest value of y

is 0 and the highest value of y is 1. Thus,

For x, the lowest value is 0, and

since , the limits of y is

Hence, we get


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1 2 2 1

0 0 0

2

, , .

x

y

f x y dydx f x y dxdy

1 2

0 2

,x

f x y dydx

0 1 and 2 2.x x y

0 1.y 10

2x y 1

2x y

11 2 2 2

0 2 0 0

, ,

y

x

f x y dydx f x y dxdy

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MULTIPLE INTEGRALS


Area

Theorem: If the region bounded below by and

above by and , then the area is given by

Remark: If the region bounded on the left by and

the right by and , then the area is given by


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2

1

g xb

a g x

A dydx

a x b R 1y g x

2y g x

2

1

h yd

c h y

A dxdy

c y d

R 1x h y

2x h y

MULTIPLE INTEGRALS


Ex: Find the area between

Sol:

First we note that the curves intersect at the points (0,0) and

(1,1). Then we see that

Ex: Find the area between Prepared by Dr. F.G.A

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R

2 3andy x y x

22

2

3

31

1 1

0 0

11 3 42 3

0 0

1 10

3 4 3 4

4 3 1.

12 12 12

g xb xx

x

a g x x

A dydx dydx y dx

x xx x dx

3 2x x

andy x y x

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MULTIPLE INTEGRALS


Volume

Ex: Find the volume of the region bounded above by

and below by

Sol:

Ex: Find the volume of the region bounded above by

and Prepared by Dr. F.G.A

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R

0 1 and 0 2.x y 2 210 3z x y

1 2

2 2 2 2

0 0

, 10 3 10 3R R

V f x y dA x y dA x y dydx

1 12

2 3 2

00 0

11

2 3

00

10 3 20 2 8 0

2 8628 2 28 .

3 3

y x y y dx x dx

x dx x x

3 0 cos , 0, , 0z c sx y x x y .y

MULTIPLE INTEGRALS

13.3: DOUBLE INTEGRALS IN POLAR COORDINATES

Let be the region bounded by the polar equations

and and the two rays and such that

For all , then the double

integral of the function

over the region is defined by

Where


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R 1r g

2r h 1 2 g h

1 2,

R

g

h

,f r

R

.dA r dr d

2

1

, ,

h

R g

f r dA f r r dr d

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MULTIPLE INTEGRALS


EX:

Sol:


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MULTIPLE INTEGRALS



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MULTIPLE INTEGRALS


Ex: Find the area inside the curve defined by

Sol:


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R

2 2sinr

2

2

0

14 8sin 4sin

2d

MULTIPLE INTEGRALS


Ex: Find the volume inside the paraboloid z = 9 − x2 − y2,

outside the cylinder x2 + y2 = 4 and above the xy-plane.

Sol:


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2 1sin 1 cos

2

2

0

14 8sin 2 1 cos2

2d

2

0

2

0

14 8sin 2 2cos2

2

1 16 8cos sin 2 12 8 0 8 6

2 2

d

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MULTIPLE INTEGRALS


Ex: evaluate where is the semicircular region

region bounded by The x-axis and the curve Prepared by Dr. F.G.A

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32 2 4

0 2

9

2 4

r rd

2 2x y

R

e dydx

R

2 2 2

0 0 0

81 81 81 56 25 2518 4

2 4 4 4 4 2d d d

21 .y x

MULTIPLE INTEGRALS

13.5: TRIPLE INTEGRALS

Def: For any function defined on the rectangular

box Q, we define the triple integral of over Q by

Where

Theorem: (Fubini’s Theorem)

Suppose that is continuous on the box Q defined by

Then, we can write the triple integral over Q as a triple

iterated integral:


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, ,f x y z

f

, ,Q

f x y z dv

.dv dAdz dxdydz

, ,f x y z

, , ,Q x y z a x b c y d and s z t

, , , ,

t d b

Q s c a

f x y z dv f x y z dxdydz

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MULTIPLE INTEGRALS


Ex: Evaluate the triple integral

where Q is the rectangular box defined by

Sol:


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, , 1 2, 0 1 0Q x y z x y and z

1 2 1

22

10 0 1 0 0

2 sin 2 sin siny y y

Q

xe z dv xe z dxdydz x e z dydz

1 1

0 0 0 0

4 sin sin 3 siny y ye z e z dydz e z dydz

11 0

00 0

3 sin 3 sinye z dz e e zdz

MULTIPLE INTEGRALS


Ex: Evaluate

Triple Integral over a Tetrahedron

Ex: Evaluate

where Q is the tetrahedron bounded by the planes x = 0,

y = 0, z = 0 and 2x + y + z = 4.


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0

0

3 1 sin 3 1 cos 3 1 1 1 6 1 .e zdz e z e e

1

0 0 1

128

xyx

xyz dzdydx

2 siny

Q

xe z dv

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MULTIPLE INTEGRALS


Sol:


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MULTIPLE INTEGRALS


تم والحمد هلل


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Documents

Calculus 2 Chapter13(6)