Double Integrals

Introduction

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 = ?

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

m

i

n

jijij AyxfV

1 1

** ),(

Definition

m

i

n

jijij AyxfV

1 1

**

nm,

),(lim

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

** ),(

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

Example 1

z=16-x2-2y2

0≤x≤20≤y≤2

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

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

Exact volume? V=48

Example 2

z

?1

]2,2[]1,1[

2

R

dAx

R

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 ),(),(

Properties

AAA

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

AA

dAyxfcdAyxcf ),(),(

AA

dAyxgdAyxf ),(),(

Linearity

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

Comparison

2121

),(),(),(AAAA

dAyxfdAyxfdAyxf

Additivity

If A1 and A2 are non-overlapping regions then

Area

AdAdAAA

of area1

A1A2

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

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

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

Note

If f (x, y) = φ (x) ψ(y) then

d

c

b

a

d

c

b

aR

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

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)