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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