Ch15(part I) (2)

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Multiple Integrals

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Geometrical Meaning

y

35

30

25

20

15

10

5

010

x5

0

-5

-10

6

42

0

-2

-4

-6

z

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Double Integrals over Rectangles

Iff(x,y) is a positive function, andRis a rectangle on thexy-

plane, then the geometrical meaning of

R dAyxf ),(

is the volume of the solid Sthat is above the rectangle R and

is below the graphz=f(x,y).

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Definition in terms of Riemann sum

Iff(x,y) is a function of two variables andR= [a,b][c,d] is

a closed rectangle in thexy-plane, then we define

AyxfdAyxf ij

m

i

n

j

ijnm

R

),(lim),( *

1 1

*

,

if the limit exists, where

the interval [a,b] is divided into mequal subintervals,

the interval [c,d] is divided into nequal subintervals,

A is the area of each sub-rectangle,

(xij*,yij*) is any point in sub-rectangle ij.

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Estimation of the integral

If we know that mf(x,y) Mon the rectangle R, then

)(area),()(area RMdAyxfRmR

Midpoint Rule

AyxfdAyxf j

m

i

n

j

i

R

),(),(1 1

wherexiis the midpoint of [xi-1,xi], andyjis the midpoint of

[yj-1,yj]

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Approximation of volume by Riemann Sum

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yx

0.50.6

0.70.8

0.91.0

0.5

2.0

1.5

1.0

-0.4-0.2

0 0.2

0.4

equation of surface z = f(x,y)

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yx

0.50.6

0.70.8

0.91.0

0.5

2.0

1.5

1.0

-0.4-0.2

0 0.2

0.4

x

equation of surface z = f(x,y)

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Iterated Integrals

Definition

The iterated integral

means that we first integrate the functionfwith respect toy,

treatingxas a constant, and then integrate the result with respect

tox. This can also be called iterated partial integration.

b

a

d

c

dxdyyxf ]),([

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Fubinis Theorem

If fis continuous on the rectangleR= [a,b][c,d], then

d

c

b

a

b

a

d

cR

dxdyyxfdydxyxfdAyxf ),(),(),(

More generally, this is true iffis bounded onR,f is

discontinuous only on a finite number of smooth curves, and

the iterated integrals exist.

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Properties of Double Integrals

IfDis the non-overlapping union of two regionsD1andD2, then

21

)()()(

DDD

dAy,xfdAy,xfdAy,xf

IfR= [a,b][c,d] is a rectangle andf(x,y) =g(x)h(y) is a

product of two functions of one variable, then

b

a

d

cR

dyyhdxxgdAy,xf

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Over General Regions

y-simple (or type I) regionsA plane regionDis said to bey-simple if it lies between the

graphs of two continuous functions ofx.

D= {(x,y): ax b,g1(x) yg2(x)}

Example:

x

1.510.50

-0.5-1-1.5

3

2.5

2

1.5

1

0.5

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In this case

b

a

xg

xgD

dydxy,xfdAy,xf2

1

)()(

x1.510.5

0-0.5-1-1.5

3

2.5

2

1.5

1

0.5

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Example

D dAyx 2Evaluate

x1.510.5

0-0.5-1-1.5

3

2.5

2

1.5

1

0.5

2

1 xy

2

2xy

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x-simple (or type II) regions

A plane regionDis said to bex-simple if it lies between the

graphs of two continuous functions ofy.

D= {(x,y): cx d, h1(y) x h2(y)}

Example:

x 543210-1-2-3

4

2

-2

-4

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In this case

d

c

yh

yhD

dxdyy,xfdAy,xf2

1)()(

x5

4

3

2

1

0-1

-2

-3

4

2

-2

-4

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Example

D

dAxyEvaluate

x5

4

3

2

1

0-1

-2

-3

4

2

-2

-4

622 xy

1xy

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Non-simple Regions

There are many regions that do not belong to any of the twoprevious types, such as the one indicated below.

In this case, we have to divide to

region into several disjoint sub-

regions so that each one is either

x-simple ory-simple.

The integral will then be the sum

of several integrals, as stated in thenext theorem.

1086420

12

10

8

6

4

2

0

D1

D2

D3

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1086420

12

10

8

6

4

2

0

Example: Evaluate D

xydA

2585 22 yx

488 22 yx

4)5(9

2 2 yx

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In Polar Coordinates

HencedA= rdrd

3

2.5

2

1.5

1

0.5

0

32.521.510.50

r= b

r= a

dr

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drdrsinr,cosrfdAy,xf

b

aR

)(

Iff is continuous on a polar rectangleRgiven by

0 a r b, , where0 - 2

then

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drdrsinr,cosrfdAy,xf

h

hD

2

1

)(

Iff is continuous on a polar regionDof the form

0 h1() r h2(),

, where0 - 2

then

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Surface Area

If Sis a surface given by the continuously differentiable

functionf(x,y) over a regionD, then its area is

D

dAyf

xfS 1Area

22

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Parametric Surface

All surfaces we have seen so far can be described by equationsof the form z=f (x,y), hence the surface can pass the vertical

line test.

In practice, many useful surfaces will not pass this test, and

will not pass any horizontal line test either.

In those cases, we cannot expect to describe the surface by

functions of the formx=g(y,z) ory= h(x,z) etc.

And the remedy is to introduce two new variables uand vthat

will control values ofx,y, andz.

The more precise definition is on the next page.

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Parametric Surface

Suppose that

vuzvuyvuxvu ,,,,,, r

is a vector-valued function defined on a regionDin the

uv-plane. The set of all points (x,y,z) in 3such that

x=x(u,v), y=y(u,v), z=z(u,v),

and (u,v) varies throughoutD, is called a parametric surface.

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Examples

A sphere centered at the origin with radius 4.

In principle this surface can be described by two equations

2222 16and16 yxzyxz

but the result in plotting is not satisfactory due to vertical

slopes near the edge of each hemisphere. (see next page)

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A sphere centered at the origin with radius 4.

20and0

cos4,sinsin4,cossin4

where

zyx

It will be a lot better to use spherical coordinates

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The curves that you can see on

the sphere are the grid curves.

The horizontal circles are

created by keeping fixed, for

instance

20where

8cos4

,sin8

sin4

,cos8

sin4

z

y

x

The vertical circles are created by

keeping fixed.

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Example 2 A parallelogram with vertexP0and two other

corners A(a1, a2, a3) and B(b1, b2, b3).

),,( 3210 cccPwhere

)()(),(

)()(),(

)()(),(

33333

22222

11111

cbvcaucvuz

cbvcaucvuy

cbvcaucvux

1,0 vuwhere

The parametric equations are

P0

A

B

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In many practical situations, we need to create the

parametric equations for a given surface, and one useful

method is to decide which directions the grid curves go first,

and then determine what parameters to use. Finally create the

equation for the surface (by adding vectors).

This can be illustrated in the next example.

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The parametric equations for the previous Torus are

x(, ) = (b+ acos )cos

y(, ) = (b+ acos )sin

z(, ) = asin

where 0 , 2

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Example 3 An ellipsoid

The standard equation in rectangular coordinates is

12

2

2

2

2

2

c

z

b

y

a

x

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1

0.5

0

-0.5

-1

-2-1

0

1

2

10

-1

x

y

One parameter will be anglewhich starts from 0 to .

Hencex= acos

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1

0.5

0

-0.5

-1

-2-1

0

1

2

10

-1

x

y

Another parameter will be anglewhich starts from 0 to 2.

For this ellipse, the

y-max = bsin and

z-max = csin

Hence

y= (bsin) cos andz= (csin) sin

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Example 3 An ellipsoid

In parametric form it can be (but not uniquely)

sinsin,cossin,cos czbyax

20,0 where

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In some extreme cases, a surface will intersect itself. If this

happens, parametric equation is the only way to describe

such as surface.

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Example 5 A surface that intersects itself

1,02,, 22 vuwherevuzvyux

4

2

0

-2

-4

1.41.210.80.60.40.20

4

3

2

1

0

E i M t h th f ll i h ith th ti

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yx

z

x

z

y

yx

z

3

Exercise: Match the following graphs with the equations on

the next slide.

I II III

IV V VI

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11. r(u,v) = cosvi+ sinvj+ uk

12. r(u,v) = ucos vi+ usin vj+ uk

13. r(u,v) = u cos vi+ u sin vj+ vk

14. r(u,v) = u3

i+ u sin vj+ u

cos

vk

15. x= (usinu)cosv,y= (1 cosu)sinv,z= u

16. x= (1u)(3 + cosv)cos4u,

y= (1u)(3 + cosv)sin4u,

z= 3u+ (1u)sinv

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Surface area of

Parametric Surfaces

D

vu dArrS Area

v

z

v

y

v

xr

u

z

u

y

u

xr vu

,,and,,where

If a smooth parametric surface S is given by the equations

Dvuvuzvuyvuxvu ),(),(),,(),,(),(r

and S is covered only once as (u,v) ranges throughout thedomainD, then the surface area is