Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d]...

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Double Integrals Introduction

Transcript of Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d]...

Page 1: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Double Integrals

Introduction

Page 2: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Volume and Double Integral

z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d]

S={(x,y,z) in R3 | 0 ≤ z ≤ f(x,y), (x,y) in R}

Volume of S = ?

Page 3: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.
Page 4: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Volume of ij’s column: Ayxf ijij ),( **

m

i

n

jijij Ayxf

1 1

** ),(Total volume of all columns:

ij’s column:

Area of Rij is Δ A = Δ x Δ y

f (xij*, yij

*)

Δ y Δ xxy

z

Rij

(xi, yj)

Sample point (xij*, yij

*)x

y

Page 5: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

m

i

n

jijij AyxfV

1 1

** ),(

Definition

m

i

n

jijij AyxfV

1 1

**

nm,

),(lim

Page 6: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Definition:

The double integral of f over the rectangle R is

if the limit exists

R

dAyxf ),(

m

i

n

jijij

R

AyxfdAyxf1 1

**

nm,

),(),( lim

Double Riemann sum:

m

i

n

jijij Ayxf

1 1

** ),(

Page 7: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Note 1. If f is continuous then the limit exists and the integral is defined

Note 2. The definition of double integral does not depend on the choice of sample points

If the sample points are upper right-hand corners then

m

i

n

jji

R

AyxfdAyxf1 1nm,

),(),( lim

Page 8: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Example 1

z=16-x2-2y2

0≤x≤20≤y≤2

Estimate the volume of the solid above the square and below the graph

Page 9: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

m=n=4 m=n=8 m=n=16V≈41.5 V≈44.875 V≈46.46875

Exact volume? V=48

Page 10: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Example 2

z

?1

]2,2[]1,1[

2

R

dAx

R

Page 11: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Integrals over arbitrary regions

A

R

f (x,y)

0

• A is a bounded plane region

• f (x,y) is defined on A• Find a rectangle R

containing A• Define new function on R:

otherwise ,0

),( if ),(),(

Ayxyxfyxf

RA

dAyxfdAyxf ),(),(

Page 12: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Properties

AAA

dAyxgdAyxfdAyxgyxf ),(),()],(),([

AA

dAyxfcdAyxcf ),(),(

AA

dAyxgdAyxf ),(),(

Linearity

If f(x,y)≥g(x,y) for all (x,y) in R, then

Comparison

Page 13: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

2121

),(),(),(AAAA

dAyxfdAyxfdAyxf

Additivity

If A1 and A2 are non-overlapping regions then

Area

AdAdAAA

of area1

A1A2

Page 14: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Computation• If f (x,y) is continuous on rectangle R=[a,b]×[c,d]

then double integral is equal to iterated integral

a bx

y

c

d

x

y

b

a

d

c

d

c

b

aR

dydxyxfdxdyyxfdAyxf ),(),(),(

fixed fixed

Page 15: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

More general case• If f (x,y) is continuous on

A={(x,y) | x in [a,b] and h (x) ≤ y ≤ g (x)} then double integral is equal to iterated integral

a bx

y

h(x)

g(x)

x

b

a

xg

xhA

dydxyxfdAyxf)(

)(

),(),(

A

Page 16: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Similarly• If f (x,y) is continuous on

A={(x,y) | y in [c,d] and h (y) ≤ x ≤ g (y)} then double integral is equal to iterated integral

d

x

y

d

c

yg

yhR

dxdyyxfdAyxf)(

)(

),(),(

c

h(y) g(y)y

A

Page 17: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Note

If f (x, y) = φ (x) ψ(y) then

d

c

b

a

d

c

b

aR

dyydxxdxdyyxdAyxf )()()()(),(

Page 18: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume.

Examples

],2/[]1,2/1[ ,)sin( AdAxyyR

2

A

x dAe where A is a triangle with vertices(0,0), (1,0) and (1,1)