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Topics to be discussed: Double integration

In rectangular coordinates, In polar coordinates Triple integration

In rectangular coordinates,In cylindricalcoordinates, In spherical coordinates Applications

Area, Volume

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4.1 Double Integral in Rectangular CoordinatesLet R be a region in the plane which is bounded by

and , where andx a,x b,y c y d a b

c d.

x a

x b

y d

y c

iRArea i ix y i i ,

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Double Integral of f over RIn symbols,

1

n

i i i in

i

lim f , x y

R

f x ,y dA

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Double Integral of f over RREMARKS:

Double integrals have the same kind of domain

additivity property that single integrals have.

Double integrals are evaluated as iterated

integrals.

dA dx dy dy dx

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Evaluating Double Integrals

2 1

2 0

4 xy dxdy

2 1

2 0

4 xy dx dy

12

2

2 0

14

2x x y dy

2

2

42

ydy

2

2

2

44

yy

8 1 8 1 16

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Region of Integration

2 1

2 0

4 xy dxdy

0x 1x

2y

2y

1 2

0 2

4 xy dydx

16

x

y

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Evaluating Double Integrals

0 0

x sinxdydx

x

2

0 0

x sinxdy dx

x

0 0

x sinxy dx

x

0

sinxdx

0

cosx

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Region of Integration

0x x

0y

y x0 0

x sinxdydx

x

2sinx

dxdyx

y

0

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Evaluating Double Integrals

3

8 2

40 1x

dydx

y

0x

3y x

8x

2y

41

dxdy

y

0

3y

0

2

x

y

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Evaluating Double Integrals3

2

40 0 1

y dxdy

y 3

2

40 0 1

y dxdy

y

32

40 1

ydy

y

2

4

0

11

4ln y

1

174

ln

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Exercise. Sketch the region of integration. Write anintegral with the order of integrationreversed. Then evaluate both integrals.

2 4

0 0

2

xxdydx

1.

2

1

0

x

xxdydx

2.

4 4 2

0 4

y /

y dxdy

3.

2

2

3 2 9 4

0 9 4

/ y

y

ydxdy

4.

32

3

435

4

3

9

2

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Application of Double IntegralsREMARKS:

If for all in a region R , thedouble integral of fover R is the volume of thesolid whose base is R and whose height at a point

in R is .

If , the double integral of fover a

region R is just the area of R .

0 f x, y x, y

x, y f x, y

1 f x, y

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x

y

z

1

n

i i i in

i

lim f , x y

z f x, y

V R

f x , y dA

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Exercise. SET-UP the double integral which givesthe volume of the solid described.a. Solid in the first octant bounded by

2 24 f x,y x y

2 24V x y dx dy 0

24 y

0

2

2 24x y

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Exercise. SET-UP the double integral which givesthe volume of the solid described.a. Solid in the first octant bounded by

2 24 f x,y x y

2 24V x y dy dx 0

24 x

0

2

2 24x y

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Exercise. SET-UP the double integral which givesthe volume of the solid described.

x

y

1

x

y x

b. Prism whose base is the

triangle in the xy-plane

bounded by the x-axis,

and the lines

and whose top lies in the

plane 3 f x,y x y

1x ,y x

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Exercise. SET-UP the double integral which givesthe volume of the solid described.

x

y

1

x

y x

3V x y dy dx 0

x

0

1

3V x y dx dy y1

0

1

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Exercise. SET-UP the double integral which givesthe volume of the solid described.c. Bounded above by

but bounded below by

2 22 f x,y x y

2 2 f x,y x y

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Exercise. SET-UP the double integral which givesthe volume of the solid described.

2 21

x y

2 22V x y dx dy 0

21 y

0

1

2 2x y dx dy 00

1

2

1 y

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Recall:

In polar form, a

point has coordinates

(r, ),

where r is the directed

distance of the point fromthe pole and is theradian measure of theangle which the terminal

side of makes with thepositive side of thex-axis,also known as the polaraxis.

(r, )

r

x= rcos

y= rsin

x2 + y2 = r2

22