let fcx) be a function defined on an
interval [a.b)
let fcx) be a function defined on an
interval [a.b)
let fcx) be a function defined on an
interval [a.b)
tsample point
What if Lex ,y ) is a function of TWO variables?
Instead of an interval [a. b) ,
we have a rectangle [a.b) x [c. d]
volume of solid
#
What if Lex ,y ) is a function of TWO variables?
Instead of an interval [a. b) ,
we have a rectangle [a.b) x [c. d]
volume of solid
and below µ,, ?,µ ⇐
rectangle R is double Riemann sum
L
if this limit exists ,
we call f INTEGRABLE
In the definition of
St Lex .y) da - m.ti.in?4.I;fCxi.s.yi.j)DA we can choose Cxij ,y¥j) = (Xi
.; , y;;) to be
sub - rectangle Rio. ( analogue of a right Riemann sum )
¥ = DA
then we can
In general , we are computing signed volume
a- DA
that lies above the square R= 51,37×52,47
and below the function fcx ,y)=e×+Y "
by
the sample point to be the upper right - corner
example : R= 51,37×52,47
((x.y) = exty '
4 equal squares the sample point to be the upper right - corner
Ans :
((x.y) = ex -' y '
4 equal squares the sample point to be the upper right - corner
Ans ! 4 Ve fcz ,3)OA t fc2.tt)dA 1- fC3 ,3) DA tf3.4) DA
Z
= e
R= { (x.y) : lxls 2 , lyle 33 and
(Cx ,y)= ¥2
R= { (x.y) : lxls 2 , lyle 33 and
'
right - hand corner of the sub - rectangle Ri; to
be the sample point , choose the middle ( center
Upper right -hand corner Midpoint
4 4
2 2
tf x -
using 4 rectangles of equal size
Ans :
tf x -
using 4 rectangles of equal size c
Ans : ve µ 's , 3) dat @E. 5) DA
+
= (I, -15312-1 ( ' -tf)2
fix) of a single variable with
b
AVERAGE VALUE of the function on Ea,b]
÷a lab dx is the AVERAGE VALUE
of the function on [a,b]
height of the
the AVERAGE VALUE of f on R is
Areatr) # (Cx .y)dA
a!Tr, # Clay )dA
Properties of double integrals :
µ fcxiyltgcx ,y) DA = ft fcx ,y) DA t fnfgcxiy) DA
tf cfcx ,y) DA = c ft fcx ,y) DA
if ex ,y) zgcxy) on R ,
then