Calculus (I)
WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
2008
WEN-CHINGLIEN Calculus (I)
16.3 Evaluating Double Integrals
(1)
Theorem
Let z =f(x,y)be a continuous function on the region R (i) If there exist continuous function x =g1(y)and x =g2(y)for which
R ={(x,y)|g1(y)≤x ≤g2(y), x ≤y ≤d}
then
d g (y) !
16.3 Evaluating Double Integrals
(1)
Theorem
Let z =f(x,y)be a continuous function on the region R (i) If there exist continuous function x =g1(y)and x =g2(y)for which
R ={(x,y)|g1(y)≤x ≤g2(y), x ≤y ≤d}
then Z Z
R
f(x,y)dA= Z d
c
Z g2(y)
g1(y)
f(x,y)dx
! dy
WEN-CHINGLIEN Calculus (I)
16.3 Evaluating Double Integrals
(1)
Theorem
Let z =f(x,y)be a continuous function on the region R (i) If there exist continuous function x =g1(y)and x =g2(y)for which
R ={(x,y)|g1(y)≤x ≤g2(y), x ≤y ≤d}
then
d g (y) !
16.3 Evaluating Double Integrals
(1)
Theorem
Let z =f(x,y)be a continuous function on the region R (i) If there exist continuous function x =g1(y)and x =g2(y)for which
R ={(x,y)|g1(y)≤x ≤g2(y), x ≤y ≤d}
then Z Z
R
f(x,y)dA= Z d
c
Z g2(y)
g1(y)
f(x,y)dx
! dy
WEN-CHINGLIEN Calculus (I)
16.3 Evaluating Double Integrals
(1)
Theorem
Let z =f(x,y)be a continuous function on the region R (i) If there exist continuous function x =g1(y)and x =g2(y)for which
R ={(x,y)|g1(y)≤x ≤g2(y), x ≤y ≤d}
then
d g (y) !
16.3 Evaluating Double Integrals
(1)
Theorem
Let z =f(x,y)be a continuous function on the region R (i) If there exist continuous function x =g1(y)and x =g2(y)for which
R ={(x,y)|g1(y)≤x ≤g2(y), x ≤y ≤d}
then Z Z
R
f(x,y)dA= Z d
c
Z g2(y)
g1(y)
f(x,y)dx
! dy
WEN-CHINGLIEN Calculus (I)
16.3 Evaluating Double Integrals
(1)
Theorem
Let z =f(x,y)be a continuous function on the region R (i) If there exist continuous function x =g1(y)and x =g2(y)for which
R ={(x,y)|g1(y)≤x ≤g2(y), x ≤y ≤d}
then
d g (y) !
Theorem
(ii) If there exist continuous functions y =h1(x)and y =h2(x)for which
R ={(x,y)|a≤x ≤b, h1(x)≤y ≤h2(x)}
then Z Z
R
f(x,y)dA= Z b
a
Z h2(x)
h1(x)
f(x,y)dy
! dx
WEN-CHINGLIEN Calculus (I)
Theorem
(ii) If there exist continuous functions y =h1(x)and y =h2(x)for which
R ={(x,y)|a≤x ≤b, h1(x)≤y ≤h2(x)}
then Z Z
R
f(x,y)dA= Z b
a
Z h2(x)
h1(x)
f(x,y)dy
! dx
Theorem
(ii) If there exist continuous functions y =h1(x)and y =h2(x)for which
R ={(x,y)|a≤x ≤b, h1(x)≤y ≤h2(x)}
then Z Z
R
f(x,y)dA= Z b
a
Z h2(x)
h1(x)
f(x,y)dy
! dx
WEN-CHINGLIEN Calculus (I)
Theorem
(ii) If there exist continuous functions y =h1(x)and y =h2(x)for which
R ={(x,y)|a≤x ≤b, h1(x)≤y ≤h2(x)}
then Z Z
R
f(x,y)dA= Z b
a
Z h2(x)
h1(x)
f(x,y)dy
! dx
Theorem
(ii) If there exist continuous functions y =h1(x)and y =h2(x)for which
R ={(x,y)|a≤x ≤b, h1(x)≤y ≤h2(x)}
then Z Z
R
f(x,y)dA= Z b
a
Z h2(x)
h1(x)
f(x,y)dy
! dx
WEN-CHINGLIEN Calculus (I)
Theorem
(ii) If there exist continuous functions y =h1(x)and y =h2(x)for which
R ={(x,y)|a≤x ≤b, h1(x)≤y ≤h2(x)}
then Z Z
R
f(x,y)dA= Z b
a
Z h2(x)
h1(x)
f(x,y)dy
! dx
Ex 2.
Z 4
0
Z √y
1
(x +y)dx
! dy
Ex 3.
Z 4
2
Z 3
1
(2x +y)dy
dx
Ex 4.
Z Z
R
2xydA
WEN-CHINGLIEN Calculus (I)
Ex 2.
Z 4
0
Z √y
1
(x +y)dx
! dy
Ex 3.
Z 4
2
Z 3
1
(2x +y)dy
dx
Ex 4.
Z Z
R
2xydA
Ex 2.
Z 4
0
Z √y
1
(x +y)dx
! dy
Ex 3.
Z 4
2
Z 3
1
(2x +y)dy
dx
Ex 4.
Z Z
R
2xydA
WEN-CHINGLIEN Calculus (I)
Ex 2.
Z 4
0
Z √y
1
(x +y)dx
! dy
Ex 3.
Z 4
2
Z 3
1
(2x +y)dy
dx
Ex 4.
Z Z
R
2xydA
Example 1:
Z Z
Ω
√x−y dxdy
WEN-CHINGLIEN Calculus (I)
Example 1:
Z Z
Ω
√x−y dxdy
Example 2:
Use the double integral to calculate the area of the regionΩenclosed by
y =x2 and x+y =2
WEN-CHINGLIEN Calculus (I)
Example 2:
Use the double integral to calculate the area of the regionΩenclosed by
y =x2 and x+y =2
Example 2:
Use the double integral to calculate the area of the regionΩenclosed by
y =x2 and x+y =2
WEN-CHINGLIEN Calculus (I)
Example 2:
Use the double integral to calculate the area of the regionΩenclosed by
y =x2 and x+y =2
Example 3:
Let R be rectanglea≤x ≤b,c ≤y ≤d show that if f is continuous on[a,b], andg is continuous on[c,d],
then Z Z
R
f(x)·g(y)dxdy = Z b
a
f(x)dx
Z d
c
g(y)dy
WEN-CHINGLIEN Calculus (I)
Example 3:
Let R be rectanglea≤x ≤b,c ≤y ≤d show that if f is continuous on[a,b], andg is continuous on[c,d],
then Z Z
R
f(x)·g(y)dxdy = Z b
a
f(x)dx
Z d
c
g(y)dy
Example 3:
Let R be rectanglea≤x ≤b,c ≤y ≤d show that if f is continuous on[a,b], andg is continuous on[c,d],
then Z Z
R
f(x)·g(y)dxdy = Z b
a
f(x)dx
Z d
c
g(y)dy
WEN-CHINGLIEN Calculus (I)
Example 3:
Let R be rectanglea≤x ≤b,c ≤y ≤d show that if f is continuous on[a,b], andg is continuous on[c,d],
then Z Z
R
f(x)·g(y)dxdy = Z b
a
f(x)dx
Z d
c
g(y)dy
Example 3:
Let R be rectanglea≤x ≤b,c ≤y ≤d show that if f is continuous on[a,b], andg is continuous on[c,d],
then Z Z
R
f(x)·g(y)dxdy = Z b
a
f(x)dx
Z d
c
g(y)dy
WEN-CHINGLIEN Calculus (I)
Example 4:
Z Z
Ω
√xy dxdy, Ω :0≤y ≤1, y2≤x ≤y
Example 4:
Z Z
Ω
√xy dxdy, Ω :0≤y ≤1, y2≤x ≤y
WEN-CHINGLIEN Calculus (I)
(2)Interchange the order of integration
For example, the iterated integral
Z 1
0
Z 1
y2
yex2dxdy
(2)Interchange the order of integration
For example, the iterated integral
Z 1
0
Z 1
y2
yex2dxdy
WEN-CHINGLIEN Calculus (I)
(2)Interchange the order of integration
For example, the iterated integral
Z 1
0
Z 1
y2
yex2dxdy
Solution:
reverse the order of integration
consider Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy
WEN-CHINGLIEN Calculus (I)
Solution:
reverse the order of integration
consider Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy
Solution:
reverse the order of integration
consider Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy
WEN-CHINGLIEN Calculus (I)
Step 1:
Identify the regionQ for which the iterated integral can be written as the double integral
Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy = Z Z
Q
f(x,y)dA
Step 1:
Identify the regionQ for which the iterated integral can be written as the double integral
Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy = Z Z
Q
f(x,y)dA
WEN-CHINGLIEN Calculus (I)
Step 1:
Identify the regionQ for which the iterated integral can be written as the double integral
Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy = Z Z
Q
f(x,y)dA
Step 1:
Identify the regionQ for which the iterated integral can be written as the double integral
Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy = Z Z
Q
f(x,y)dA
WEN-CHINGLIEN Calculus (I)
Step 2:
Find constantsaandband continuous functions g1and g2 so that the region Q can be expressed as
Q={(x,y)|a≤x ≤b, g1(x)≤y ≤g2(x)}
Step 2:
Find constantsaandband continuous functions g1and g2 so that the region Q can be expressed as
Q={(x,y)|a≤x ≤b, g1(x)≤y ≤g2(x)}
WEN-CHINGLIEN Calculus (I)
Step 2:
Find constantsaandband continuous functions g1and g2 so that the region Q can be expressed as
Q={(x,y)|a≤x ≤b, g1(x)≤y ≤g2(x)}
Step 2:
Find constantsaandband continuous functions g1and g2 so that the region Q can be expressed as
Q={(x,y)|a≤x ≤b, g1(x)≤y ≤g2(x)}
WEN-CHINGLIEN Calculus (I)
Step 3:
Rewrite the iterated integral as Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy = Z Z
Q
f(x,y)dA
= Z b
a
Z g2(x)
g1(x)
f(x,y)dydx
Step 3:
Rewrite the iterated integral as Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy = Z Z
Q
f(x,y)dA
= Z b
a
Z g2(x)
g1(x)
f(x,y)dydx
WEN-CHINGLIEN Calculus (I)
Step 3:
Rewrite the iterated integral as Z d
c
Z h2(y)
h1(y)
f(x,y)dxdy = Z Z
Q
f(x,y)dA
= Z b
a
Z g2(x)
g1(x)
f(x,y)dydx
Example 1:
Z 1
0
Z 1
y2
yex2dxdy
Example 2:
Z 1
0
Z
√1−x
0
xy2dydx
WEN-CHINGLIEN Calculus (I)
Example 1:
Z 1
0
Z 1
y2
yex2dxdy
Example 2:
Z 1
0
Z
√1−x
0
xy2dydx
Example 1:
Z 1
0
Z 1
y2
yex2dxdy
Example 2:
Z 1
0
Z
√1−x
0
xy2dydx
WEN-CHINGLIEN Calculus (I)
Example 1:
Z 1
0
Z 1
y2
yex2dxdy
Example 2:
Z 1
0
Z
√1−x
0
xy2dydx
Example 3:
1 Z 2
0
Z x2
0
(x −2y)dydx
2 Z 1
0
Z 1
x2
xey2dydx
WEN-CHINGLIEN Calculus (I)
Example 3:
1 Z 2
0
Z x2
0
(x −2y)dydx
2 Z 1
0
Z 1
x2
xey2dydx
Example 3:
1 Z 2
0
Z x2
0
(x −2y)dydx
2 Z 1
0
Z 1
x2
xey2dydx
WEN-CHINGLIEN Calculus (I)
Thank you.