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Warm-Up: Double Integrals
Graph the curves (sketch them on paper!):
y = x2 and y =√
x
Enter 1 when you have finished. Enter 0 if you run out of
time. Enter 500 if you do not know how to answer this
question.
Warm-Up: Finding Limits of IntegrationFind the limits of integration for the double integral:
¨D
f (x , y ) dA
where D is the region in the xy -plane bounded by the
curves:
y = x2 and y =√
x
using the order of integration:
dA = dy dx .
Enter 1 when you have finished. Enter 0 if you run out of
time. Enter 500 if you do not know how to answer this
question.
Warm-Up: Double Integrals
The lower limit of integration for the “inside” integral is:
A. x = y 2.
B. y = x2.
C. x =√
y .
D. y =√
x .
E. None of the above.
Warm-Up: Double Integrals
The upper limit of integration for the “inside” integral is:
A. x = y 2.
B. y = x2.
C. x =√
y .
D. y =√
x .
E. None of the above.
Warm-Up: Double Integrals
The lower limit of integration for the “outside” integral is:
A. x = y 2.
B. y = x2.
C. x =√
y .
D. y =√
x .
E. None of the above.
Warm-Up: Double Integrals
The lower limit of integration for the “outside” integral is:
Enter your answer as a number. If you are not sure, enter
500,
Warm-Up: Double Integrals
The upper limit of integration for the “outside” integral is:
Enter your answer as a number. If you are not sure, enter
500,
“Reading” a Double Integral
¨D
f dA
I D is the region (or domain) of integration. D ⊆ R2.
I f , a function of two variables, is called the integrand. f
must be defined over D.
I dA is the area element — it can be thought of as
representing the area of an infinitesimal (very very
small) region in R2.
Application of Integrals: Motivation
Integrating is a process of “chopping and adding”:
¨D
f (x , y ) dA
Double Integrals & Area
Set up the double integral:
¨D
f (x , y ) dA
where D is the region in the xy -plane bounded by the
curves:
y = x2 and y =√
x .
and f (x , y ) = 1.
Double Integrals & Area
What is the approximate area of the region D?
Enter your answer as a number. If you are not sure, enter
500.
An Application of Double Integrals: Area
The area AD of a planar region D is given by the double
integral:
AD =
¨D
1 dA =
¨D
dA.
I Chop up region D into infinitesimal rectangles, each of
area dA = dx dy = dy dx .
I Integration adds up the areas of all of the infinitesimal
rectangles.
Example: Double Integrals and Area
Suppose D is the region in the first quadrant, between the
two circles x2 + y 2 = 1 and x2 + y 2 = 4.
I Set up a double integral
¨D
dA that gives the area of
D.
Double Integrals & Area
The minimum number of double integrals you will need to
evaluate in order to compute the area of the region:
D : 1 ≤ x2 + y 2 ≤ 4, x , y ≥ 0
in Cartesian coordinates is:
Enter your answer as a number. If you are not sure, enter
500.
Double Integrals & Area
What integration technique will you need to use to evaluate
this area integral
A. u-sub.
B. trig sub.
C. integration by parts.
D. None of the above.
This has been a public service announcement, brought to
you by polar coordinates.
Double Integrals & Area
What integration technique will you need to use to evaluate
this area integral
A. u-sub.
B. trig sub.
C. integration by parts.
D. None of the above.
This has been a public service announcement, brought to
you by polar coordinates.
An Application of Double Integrals: VolumeSuppose f (x , y ) ≥ 0 over a planar region D.
The volume of the three-dimensional region between D and
the graph of f (x , y ) is given by the double integral:
V =
¨D
f (x , y ) dA
I Chop up region D into infinitesimal rectangles, each of
area dA = dx dy = dy dx .
I f (x , y ) dA is the volume of a box with height f (x , y )
and an infinitesimal base of area dA.
I Integration adds up the volumes of all of the
infinitesimal boxes.
An Application of Double Integrals: MassThe mass m of a flat object occupying a planar region D is
given by the double integral:
m =
¨D
σ(x , y ) dA
where σ(x , y ) ≥ 0 gives the (surface) density of the object
at each point (x , y ) in the region D.
I Chop up region D into infinitesimal rectangles, each of
area dA = dx dy = dy dx .
I Since mass is density times area, the mass of an
infinitesimal rectangle is: dm = σ(x , y ) dA.
I Integration adds up the areas of all of the infinitesimal
rectangles.
Clicker Question
I will always sketch the region of integration D.
(Even if it seems like I don’t need to.)
A. Yes, I will sketch the region D.
B. Yes, I promise I will sketch the region D.
C. If I don’t sketch the region D, and I do badly on the
next exam, I understand that Dr. Ultman will write “I
told you so”.