Post on 06-Aug-2015
Maths-I (calcules)-2110014
Presentation on:double integral
Example 1: A Double Integral Over a Rectangular Region
0 2
sin(y) cos(x) 1 dy dx
y
y 20
cos(y) ycos(x) y dx
0
2 3 3 cos(x d) x
02x 3 x 3 sin(x)
22 3
R
Let f(x,y) sin(y) cos(x) 1andletR be the region in the xy-plane bounded
by x 0,x ,y 2 ,and y .Calculate f(x,y)dA :
is a function that sweeps out
vertical area cross sections f
2 3 3 cos(x)
romx 0to x .
If weintegrate these areacross sections,
we accumulatethe volume of the solid:
Example 2: Changing the Order of Integration
2 0
sin(y) cos(x) 1 dx dy
x
x 02
xsin(y) sin(x) x dy
2
sin(y dy)
2
y cos(y)
22 3
R R R
Calculate f(x,y)dA by computing f(x,y)dy dx insteadoff (x,y)dx dy.
is a function that sweeps out
horizontal area cross sections fromy 2
to y . If weintegrate these areacross
sections,we accumulatethe volume of the solid:
sin(y)
Computing Double Integrals Over a Rectangular Region
d b
c a
1)Setup f(x,y)dx dy.
d dx b
x ac c
2)Hold y constant and
integrate with respect to x:
F(x,y) dy F(b,y) F(a,y) dy
R
Let z f(x,y)beasurfaceandletR be the rectangular region in the
xy-plane boundedby a x bandc y d.Calculate f(x,y)dA :
d
c
3)F(b,y) F(a,y) isafunctionof y
thatmeasuresthe area of horizontal
crosssections between y c and y d.
So we can integrate these area cross
sectionsto get our volume:
F(b,y) F(a,y) dy
b d
a c
1)Setup f(x,y)dy dx.
b bx d
x ca a
2)Hold x constant and
integrate with respect to y:
G(x,y) dx G(x,d) G(x,c) dx
b
a
3)G(x,d) G(x,c) isafunctionof x
thatmeasuresthe area of vertical
crosssections between x a and x b.
So we can integrate these area cross
sectionsto get our volume:
G(x,d) G(x,c) dx
Example 3: Computing Another Double Integral Over a Rectangular Region
3 4
3 4
2sin(xy) 1 dx dy
x 43
3 x 4
2cos(xy)x dy
y
3
3
2cos(4y) 2cos( 4y)4 ( 4) dy
y y
3
3
8dy
y 3
y 38y
48
R
Let f(x,y) 2sin(xy) 1andletR be the region in the xy-plane boundedby
x 4,x 4,y 3,and y 3.Calculate f(x,y)dA :
How did this end up negative??
Example 4: A Double Integral Over a Non-Rectangular Region
2 2
2
R
Let f(x,y) x y 3andletR be the region in the xy-plane boundedabove by
y x 4andbelowby y 5.Calculate f(x,y)dA :
2y3 32
3 y 5
x 4y
yx 3y dx3
62 4
3
3
x90 10x 3x dx
3
R
f(x,y)dA
highright
left lowy (x)
y (x)x
2 2
x
x y 3 dy dx 2highy (x) x 4
lowy (x) 5leftx 3 rightx 3
2
2 23
5
4
3
x
x y 3 dy dx
62 4 x
90 10x 3x is a function that sweeps 3
out vertical area cross sections from 3to3.
62 4
If weintegrate these cross sections given by
x90 10x 3x fromx 3to x 3,
3we accumulatethe volume of the solid:
Example 4: A Double Integral Over a Non-Rectangular Region
2 2
2
R
Let f(x,y) x y 3andletR be the region in the xy-plane boundedabove by
y x 4andbelowby y 5.Calculate f(x,y)dA :
62 4
3
3
x90 10x 3x dx
3
73 5
3
3
10 3 x21
90x x x3 5
1551635
Example 5: Change the Order of Integration
R
Calculate f(x,y)dA again,but changetheorder ofi ntegration:
4 yx4
5 y
32
4x
xy x 3x dy
3
24
5
26 4 y 2y 4 y 2y 4 dyy
3 3
R
f(x,y)dA
ht ighop
bottom lowx (y)
x (y)y
y
2 2x y 3 dx dy highx (y) 4 y
lowx (y) 4 y
topy 4
2 244
4
y
5 y
x y 3 dx dy
226 4 y 2y 4 y 2y 4 y is a function
3 3that sweepsouthorizontal area cross sections
fromy 5to y 4.
bottomy 5
226 4 y 2y 4 y 2y 4 y is a function
3 3that sweepsouthorizontal area cross sections
fromy 5to y 4.
Example 5: Change the Order of Integration
2 2
2
R
Let f(x,y) x y 3andletR be the region in the xy-plane boundedabove by
y x 4andbelowby y 5.Calculate f(x,y)dA :
24
5
26 4 y 2y 4 y 2y 4 dyy
3 3
4
3/2 2
5
4(4 y) 15y 41y 261
105
1551635
Summary of Example 4 & 5
2
5 4 y
2 2 24 yx3 44
3
2
5
x y 3 dy dx and x y 3 dx dy compute
the exact same thing!!!
highx (y) 4 y lowx (y) 4 y
topy 4
bottomy 5
2highy (x) x 4
lowy (x) 5leftx 3 rightx 3
We have changed the order of integration. This works whenever
we can bound the curve with top/bottom bounding functions
or left/right bounding functions.
Computing Double Integrals Over a Non-Rectangular Region
right high
left low
x y (x)
x y (x)
1)Setup f(x,y)dy dx
right right
high
lowleft left
x xy y (x)
high lowy y (x)x x
2)Hold x constant and
integrate with respect to y:
F(x,y) dx F(x,y (x)) F(x,y (x)) dx
R
Let z f(x,y)beasurfaceandletR be the non-rectangular region in the
xy-plane boundedby y f(x)ontopand y g(x)onbottom.Calculate f(x,y)dA :
high low
left right
high
3) F(x,y (x)) F(x,y (x)) isafunction
of x thatmeasuresthe area of vertical
crosssections between x x and x x .
So we can integrate these area cross
sectionsto get our volume:
F(x,y (x)) F(x
right
left
x
lowx
,y (x)) dx
Likewisefor theorder
integration being reversed.
OldWay:
Example 6: Using the Double Integral to Find the Area of a Region
2
LetR be the regioninthe xy-plane fromExample4and5that isboundedabove
by y x 4andbelowby y 5.Find theareaof R.
3
2
3
x 4 5 dxR
New Way:
3
2
3
x 9 dx
33
3
x9x
3
36
23 x 4
3 5
1dy dx
R
1dA
23
y x 4
y 53
y dx
3
2
3
x 4 5 dx
RR
Claim:For aregionR, Area 1dA
It works!!!
Properties of Double Integrals
The two intervals determine the region of integration R on the iterated integral
The Gauss-Green Formula
high
low
t
R t
x' n x((t)n m
dA dtm x(t),y(t t),x y
y() '(t)t) y
low high
LetR be a region in the xy-plane whoseboundary is parameterized
by(x(t),y(t)) for . Thenthefollowingformulahot :t st ld
Main Idea: You can compute a double integral
as a single integral, as long as you have
a reasonable parameterization of th boundary
cu
e
rve.
low high
MainRule: For this formula to hold:
1) Your parameterization must be counterclockwise.
2) Your mustbring youaroundthe
boundary curveexactly ONCE.
3) Your mustbea sibound mple
t
c
t
l
t
oary sedcu curve.rve
22
R
Example6:Let f(x,y) y xy 9 and let R by the region in the xy-plane
yxwhoseboundary is describedby the ellipse 1.
3 4
Find f(x,y)dx dy :
Let x(t),y(t) 3cos(t),4sin(t) tfor 0 2
high
lowR
t
t
m x(t x'),y(tn m
Gauss-Green: dx dy dtx
n x(t),y(t y( 'y
() tt) ))
x
0
Set f(s,y)n ,y dsx x
0
y sy 9 ds x2
0
s ysy 9s
2
2x y
xy 9x2
y'(t)x'(t) 3sin(tFurther, , ,4co )) .s(t
high
low
t
t
n x(t),y(t y'(x'(t t)Plugin: dm x(t), t)t )y( )
2n x(t),y(t) 12sin(t)cos(t) 18cos (t)sin(t) 27S to cos( )
22
Example6:Let f(x,y) y xy 9 and let R by the region in the xy-plane
yxwhoseboundary is describedby the ellipse 1.
3 4
high
low
t
t
x' n x
Pluginto Gaus
(t),y(t y'(t(t) )m x(t),y(t
s-Green:
) dt)
22
0
12sin(t)cos(t) 18cos (t)sin d(t) 27co( 3si s(t) t(4cos(t)t )n( ))0
2
2 3 2
0
48cos (t)sin(t) 72cos (t)sin(t) 108cos (t) dt
2
low high
n x(t),y(t) 12sin(t)cos(t) 18cos (t)sin(t)
[t ,t ] [0
y'(t) 4cos(t)
x'(t) 3si
m x(t),y(t) 0
,2 ]
27cos(t)
n(t)
k 1 k 1b k
a
k 1 k 1b k
a
2
(c
(a)(
sin (b) sin a)Hint: sin (t) os(t)dt
k 1 k 1cos (b) cos
and cos in(t)dtk 1 k 1
1 cos(2t)and cos (t)
t)
2
s
108
Double Integral and Fibuni’s Theorem
The integral of an integral Another Method for finding
• Volume• Mass density• Centers of mass• Joint probability• Expected value
Fibuni’s Theorem states that if f is continuous on a plane region R
Integrate with respect to x first then y when you have a region R that is bounded on the left and right by two functions of y:
Integrate with respect to y first then x when you have a region R that is bounded on the top and bottom by two functions of x:
Use Gauss-Green when you have a region R whose boundary you can parameterize with a single function of t, (x(t),y(t)) fortlow ≤ t ≤ thigh.
When Should I Use What??
right high
left low
x y (x)
x y (x)
f(x,y)dy dx high
low
t
t
n x(t),y(t) y'(t)dt
low hight t t
(x(t),y(t))
R
hightop
bottom low
x (y)y
y x (y)
f(x,y)dy dx
highy (x)
lowy (x)
Rlowx (y)highx (y)
R