082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

880 Chapter 27 Line integrals and multiple integrals
Solutions
3 (b) φ=xy+c (c) xy+c (d) 38
4 φ=x 2 y−y 4 x+3x+c,5
5 2
6 (a) F = 4yi +4xj (b) 64 (c) 64
27.6 DOUBLEANDTRIPLEINTEGRALS
27.6.1 Doubleintegrals
Expressions such as
∫ y=y2
x=x 1
y=y 1
∫ x=x2
f(x,y)dxdy
are known asdouble integrals. What ismeant by the above is
∫ (
y=y2 ∫ )
x=x2
f(x,y)dx dy
y=y 1
x=x 1
wherefirstlytheinnerintegralisperformedbyintegrating f withrespecttox,treatingy
asifitwereaconstant.Thelimitsofintegrationareinsertedasusualandthenthewhole
expression isintegrated with respect toybetween the limitsy 1
andy 2
.
It is important that you can distinguish between double integrals and line integrals.
Physically they mean quite different things, and they are evaluated in different ways.
Whilstbothtypesofintegralcontaintermsinvolvingx,y,dxanddy,inadoubleintegral
dx and dy always occur as a product, that is as dxdy. In a line integral they occur separately.
When evaluating a line integral we integrate along a curve. When evaluating a
double integral we integrate over a two-dimensional region.
Double integrals may alsobe writteninthe form
∫ ∫
f(x,y)dxdy
R
whereRiscalledtheregionofintegration.TheregionRmustbedescribedmathematically.
Inthe integral
∫ y=y2
x=x 1
y=y 1
∫ x=x2
f(x,y)dxdy
R is the rectangular region defined by x 1
x x 2
, y 1
y y 2
. Non-rectangular
regions are alsocommon.
Example27.9 Sketch the regionRdefined by 1 x4 and 0 y2.
Solution TheregionisshowninFigure27.9.Weseethatxliesbetween1and4,andyliesbetween
0 and 2.Consequently the region isrectangular.