Double Integrals over General Regions

Math 55 - Elementary Analysis III

Institute of MathematicsUniversity of the Philippines

Diliman

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Recall

If f(x, y) is integrable over the rectangular regionR = [a, b] [c, d], then

R

f(x, y) dA =

ba

dcf(x, y) dy dx

=

dc

baf(x, y) dx dy

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Type I Region

A plane region D is said to be of Type I if it lies between twocontinuous functions of x, that is,

D = {(x, y) : a x b, g1(x) y g2(x)}

where g1 and g2 are continuous on [a, b].

a b

y = g2(x)

y = g1(x)

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Integral over Type I Region

If f is continuous on a Type I region D such that

D = {(x, y) : a x b, g1(x) y g2(x)} ,

then

D

f(x, y) dA =

ba

g2(x)g1(x)

f(x, y) dy dx

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Integral over Type I Region

Example

Evaluate

R

(2x y) dA where R is the region enclosed by

y = x2 and x = y2.

Solution. Since R is a Type I region, i.e.,

R ={(x, y) : 0 x 1, x2 y x} ,

1

1

y = x2

y =x

R

(2x y) dA = 10

xx2

(2x y) dy dx

=

10

(2xy y

2

2

) y=x

y=x2dx

=

10

[(2x

32 x

2

)(2x3 x

4

2

)]dx

=

(4x

52

5 x

2

4 x

4

2+x5

10

)10

=3

20Math 55 Double Integrals over General Regions 5/ 14

Type II Region

A plane region D is said to be of Type II if it lies between twocontinuous functions of y, that is,

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

where h1 and h2 are continuous.

x = h2(y)x = h1(y)

c

d

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Integral over Type II Region

If f is continuous on a Type II region D such that

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

then

D

f(x, y) dA =

dc

h2(y)h1(y)

f(x, y) dx dy

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Integral over Type II Region

Example

Evaluate

R

(2x y) dA where R is the region enclosed by

y = x2 and x = y2.

Solution. We consider R as a Type II region, i.e.,

R ={(x, y) : y2 x y, 0 y 1} ,

1

1

x =y

x = y2

R

(2x y) dA = 10

yy2

(2x y) dx dy

=

10

(x2 xy) x=

y

x=y2dy

=

10

[(y y 32

) (y4 y3)] dy

=

(y2

2 2y

52

5 y

5

5+y4

4

)10

=3

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Double Integrals over General Regions

Some properties

1 If c is a constant,

R

cf(x, y) dA = c

R

f(x, y) dA

2

R

[f(x, y) g(x, y)] dA =

R

f(x, y) dA

R

g(x, y) dA

3 If R = R1 R2 such that R1 R2 = , then

R

f(x, y) dA =

R1

f(x, y) dA+

R2

f(x, y) dA

Math 55 Double Integrals over General Regions 9/ 14

Double Integrals over General Regions

Example

Set up the iterated double integral equal to

R

xy dA where R

is the region bounded by y =x+ 1, x+ 2y = 2 and the x-axis

by considering R as a

a. Type I region b. Type II region

Solution.

1 1 2

1y =x + 1

y = 2x2

a. R ={(x, y) : 1 x 0, 0 y x+ 1} {

(x, y) : 0 x 2, 0 y 2x2}, hence,

R

xy dA =

01

x+10

xy dy dx+

20

2x2

0

xy dy dx

Math 55 Double Integrals over General Regions 10/ 14

Double Integrals over General Regions

Example

Set up the iterated double integral equal to

R

xy dA where R

is the region bounded by y =x+ 1, x+ 2y = 2 and the x-axis

by considering R as a

a. Type I region b. Type II region

Solution.

1 1 2

1x = y2 1 x = 2 2y

b. R ={(x, y) : y2 1 x 2 2y, 0 y 1}, hence,

R

xy dA =

10

22yy21

xy dx dy

Math 55 Double Integrals over General Regions 11/ 14

Double Integrals over General Regions

Example

Evaluate

20

2yex

2dx dy.

Solution. f(x, y) = ex2

has no elementary antiderivative withrespect to x. Consider the region R = {(x, y) : y x 2, 0 y 2}.Write R as R = {(x, y) : 0 x 2, 0 y x}.

1 2

1

2

x = y

20

2y

ex2

dx dy =

R

ex2

dA =

20

x0

ex2

dy dx

=

20

yex2

y=xy=0

dx =

20

xex2

dx

=ex

2

2

20

=e4 12

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Exercises

1 Evaluate the following iterated double integrals:

a. 20

2yy

xy dx dy

b.

pi0

piy

cosx2 dx dy

c.

40

2x

1

1 + y3dy dx

d.

10

1x

exy dy dx

2 Evaluate the following double integrals over the given region R.

a.

R

(x+ y2) dA, R is the triangle with vertices (0, 0), (1, 1), (0, 2)

b.

R

x sin y dA, R is bounded by y = x2, x = 0 and y = pi.

c.

R

x2 dA, R is bounded by y = x, 2x+ y = 6 and the y-axis.

3 Given the iterated integral

D

f(x, y) dA =

10

2y0

f(x, y) dx dy +

31

3y0

f(x, y)dx dy,

sketch the region D and express the double integral as an iterated

integral with reversed order of integration.Math 55 Double Integrals over General Regions 13/ 14

References

1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008

2 Leithold, L., The Calculus 7, Harper Collins College Div., 1995

3 Dawkins, P., Calculus 3, online notes available athttp://tutorial.math.lamar.edu/

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