Topic 3.1 (continued): Applications of

Double Integrals

Textbook: Sections 15.2, 15.4

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

Did you sketch/graph the region of integration?

A. Yes.

B. No.

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,

Big Ideas

I Applications of the Double Integral: Area, Volume, and

Mass.

“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”.