Chapter TwelveChapter Twelve

Multiple IntegralsMultiple Integrals

Section 12.1Section 12.1

Double Integrals Over Double Integrals Over

RectanglesRectangles GoalsGoals

Volumes and double integralsVolumes and double integrals Midpoint RuleMidpoint Rule Average valueAverage value Properties of double integralsProperties of double integrals

Volumes and Double Volumes and Double IntegralsIntegrals

Given a function Given a function ff (x, y), defined on a (x, y), defined on a closed rectangleclosed rectangle

Suppose: Suppose: ff((xx, , yy) ≥ 0.) ≥ 0.

Question: What is the volume of the Question: What is the volume of the solid solid SS under the graph of under the graph of ff and above and above RR??

Volumes (cont’d)Volumes (cont’d)

Volumes (cont’d)Volumes (cont’d)

We do this byWe do this by dividing the interval [dividing the interval [aa, , bb] into ] into mm

subintervals [subintervals [xxii-1-1, , xxii] of equal width ] of equal width x = x =

((bb – – aa)/)/mm and and dividing [dividing [cc, , dd] into ] into nn subintervals [ subintervals [yyjj-1-1, , yyjj] ]

of equal width of equal width y = y = ((dd – – cc)/)/nn..

Next we form the subrectanglesNext we form the subrectangles

each with area each with area AA = = xxyy : :

Volumes (cont’d)Volumes (cont’d)

Volumes (cont’d)Volumes (cont’d)

We choose a We choose a sample pointsample point Then we can approximate the part of Then we can approximate the part of SS

that lies above each that lies above each RRijij by a thin by a thin

rectangular box with base rectangular box with base RRijij and heightand height

The volume of this box is the height of The volume of this box is the height of the box times the area of the base the box times the area of the base rectangle:rectangle:

* *, in each .ij ij ijx y R

* *, as shown on the next slide.ij ijf x y

Volumes (cont’d)Volumes (cont’d)

Volumes (cont’d)Volumes (cont’d)

Following this procedure for all the Following this procedure for all the rectangles and adding the volumes of rectangles and adding the volumes of the corresponding boxes, we get an the corresponding boxes, we get an approximation to the total volume of approximation to the total volume of SS::

This is illustrated on the next slide:This is illustrated on the next slide:

* *,ij ijf x y A

* *

1 1

,m n

ij iji j

V f x y A

Volumes (cont’d)Volumes (cont’d)

Volumes (cont’d)Volumes (cont’d)

As As mm and and n n become larger and larger become larger and larger this approximation becomes better this approximation becomes better and better.and better.

Thus we would expect thatThus we would expect that

We use this expression to define the We use this expression to define the volumevolume of of SS..

* *

,1 1

lim ,m n

ij ijm ni j

V f x y A

Double IntegralDouble Integral

Limits of the preceding type occur Limits of the preceding type occur frequently in a variety of settings, so frequently in a variety of settings, so we make the following general we make the following general definition:definition:

Double Integral (cont’d)Double Integral (cont’d)

A volume can be written as a double A volume can be written as a double integral:integral:

Double Integral (cont’d)Double Integral (cont’d)

The sum in our definition of double The sum in our definition of double integral is called a integral is called a double Riemann double Riemann sumsum and is an approximation to the and is an approximation to the double integral.double integral.

If If ff happens to be a happens to be a positivepositive function, function, then the double Riemann sum is the then the double Riemann sum is the sum of volumes of columns and sum of volumes of columns and approximates the volume under the approximates the volume under the graph of graph of ff..

ExampleExample Estimate the volume of the solid that liesEstimate the volume of the solid that lies

aboveabove the square the square RR = [0, 2] = [0, 2] [0, 2] and [0, 2] and belowbelow the elliptic paraboloid the elliptic paraboloid zz = 16 – = 16 – xx22 – 2 – 2yy22..

Divide Divide RR into four equal squares and into four equal squares and choose the sample point to be the upper choose the sample point to be the upper right corner of each square right corner of each square RRijij..

Sketch the solid and the approximating Sketch the solid and the approximating rectangular boxes.rectangular boxes.

SolutionSolution

The squares are shown on the next The squares are shown on the next slide.slide.

The paraboloid is the graph ofThe paraboloid is the graph offf((xx, , yy) = 16 – ) = 16 – xx22 – 2 – 2yy22 and the area of and the area of each square is 1. Approximating the each square is 1. Approximating the volume by the Riemann sum with volume by the Riemann sum with mm = = nn = 2, we have = 2, we have

Solution (cont’d)Solution (cont’d)

Solution (cont’d)Solution (cont’d) Thus 34 is theThus 34 is the

volume of thevolume of theapproximatingapproximatingrectangular boxesrectangular boxesshown:shown:

Using More SquaresUsing More Squares

We get better approximations to the We get better approximations to the volume in the preceding example if volume in the preceding example if we increase the number of squares.we increase the number of squares.

The next slides show how the The next slides show how the columns start to look more like the columns start to look more like the actual solid when we use 16, 64, and actual solid when we use 16, 64, and 256 squares:256 squares:

Using More Squares Using More Squares (cont’d)(cont’d)

Using More Squares Using More Squares (cont’d)(cont’d)

The Midpoint RuleThe Midpoint Rule

We use a double Riemann sum to We use a double Riemann sum to approximateapproximate the double integral. the double integral.

The sample pointThe sample point

to be the to be the centercenter

chosen is in , **ijijij Ryx

: of , ijji Ryx

ExampleExample

Use the Midpoint Rule with Use the Midpoint Rule with mm = = nn = = 2 to estimate the value of2 to estimate the value of

SolutionSolution We evaluate We evaluate ff((xx, , yy) = ) = xx – 3 – 3yy22 at the centers of the four at the centers of the four subrectangles shown on the next subrectangles shown on the next slide:slide:

where,3 22 R

dAyx

.21,20|, yxyxR

Solution (cont’d)Solution (cont’d)

Solution (cont’d)Solution (cont’d) The area of each subrectangle is The area of each subrectangle is ΔΔAA = =

½, so½, so to equal elyapproximat is 3 22 R

dAyx

Using More Using More SubrectanglesSubrectangles

If we keep dividing each subrectangle into four smaller ones, we get the Midpoint Rule approximations shown.

These valuesapproach the exactvalue of the doubleintegral, –12.

Average ValueAverage Value

We define the We define the average valueaverage value of a of a function function ff of one variable defined on of one variable defined on a rectangle a rectangle RR as as

where where AA((RR) is the area of ) is the area of RR.. If If ff((xx, , yy) ≥ 0, the equation) ≥ 0, the equation

R

dAyxfRA

f ,1

ave

R

dAyxffRA ,ave

Average Value (cont’d)Average Value (cont’d)

says that the box with base says that the box with base RR and and height height ffaveave has the same volume as the has the same volume as the

solid that lies under the graph of solid that lies under the graph of ff.. If If zz = = ff((xx, , yy) describes a mountainous ) describes a mountainous

region and we chop off the tops of the region and we chop off the tops of the mountains at height mountains at height ffaveave, then we can , then we can

use them to fill in the valleys so that use them to fill in the valleys so that the region becomes completely flat:the region becomes completely flat:

Average Value (cont’d)Average Value (cont’d)

Properties of Double Properties of Double IntegralsIntegrals

On the next slide we list three On the next slide we list three properties of double integrals.properties of double integrals.

We assume that all of the integrals We assume that all of the integrals exist.exist.

The first two properties are referred The first two properties are referred to as the to as the linearitylinearity of the integral: of the integral:

Properties (cont’d)Properties (cont’d)

If If ff((xx, , yy) ≥ ) ≥ gg((xx, , yy) for all () for all (xx, , yy) in ) in RR, , thenthen

ReviewReview

Volumes and double integralsVolumes and double integrals Definition of double integral using Definition of double integral using

Riemann sumsRiemann sums Midpoint RuleMidpoint Rule Average valueAverage value Properties of double integralsProperties of double integrals