Chapter 5-Double and Triple Integrals

Chapter 5
:
Double and triple integrals


5 1 .
:
Introduction


Double Integrals as volumes :




volume :



(p(x 4) ,
da



Unf(x 3) dx dy ,




Cavalieri's principle :




A(X) :
cross-sectional area


volume
/A(Xax suce a
method
:
-




·


&A((i) (Xi +
-



Xi




Reduction to Iterated Integrals :




z =
f(x y) , defined on (a , b]x(C , d]


cross-section area :
@f f(xo , y) from y c to
:

y d
=




↓
cut wrt X-axis "
A(xo) =


( f(x0 y)dy
,




v(Xo) =


13d(X , Y) dyd


v(X) =

(f(x Y) d ,




5 2:
.
The double integral over a rectangle



him Tin)DxDy :



/Yay

Properties :




1
(p(X 3) + g(x Y()dA
((p f(x y)dA ))9(X Y)dA linearity
=
. , , , + ,




2 .




(p(f(x Y)dA c((pf(x 3) dA ,
=
,
homogeneity




.
3 When f(x , y) g(X , Y) ,
:




(p
+ (X 4)dAy, ,
//p9(X 4) dA ,
monotonicity




4 .
if Q :
ERi :




Naf(X Y)dA = , X , YIdA additivity




Fubini's Theorem :




+x , ) dyax
= x , yax a