ege let B base be a nice subset of R
let D cone in 1133 with base B on Xy plaice0,99
ay
X B Oz
Howtofindvolume of D Vol ArealBz 07
Answer by conceptof Riemann sum and this figure
Vol Soh Area de
By similarity ratio ofheight that I Eratio of the area
AreaAreaTBECEJ
Vol 5 SohCI 5ArealB At
Sohc Ejaz Area
IsAreaB Check
Cylindrical Coordinates at1133 itCr 0,21
Bo _polarcoordinates for r
the y plane µc yp crosgraagz
820
z rectangular vertical I 9coordinates t x y oX 11
Then a point P ix y z can becrosorseioso
represented by 18,0 where
I X rose
y rsinoz Z
And Cr z is called the atmdri as forIRS
Remarki cc is a constant n
KC c o describes a cylinder
µ uso c describes a fitI 4r ci sz c descries a horizontalplane i
I 1as inrectangular coordinates iz
Remake Wecan define cylindrical Xcoordinates in other directors y
h d
egi x x xi O c
1
Volumneelement
dV dxdydzrTirdotaz
orderofthe integration canbe changed L
1role
ofr dr
de
egI seealsoegatFindthe volume of the
Tiffin.ggeeIgim
m't.ae
it
Sole az µ
Ofixed µzIE i y
1 t.EE er.ofncrao.rsoaidiskof Er
Fabinis radiosE21T
Voeux f JeffgFrdzdrdof fo rut E drdo check
Spherical coordinates in IR ta quay f rho
g 4,0 where gasol ii
ig distance from theorigin geo
to angle from the pantier
Z axis to optOf loECT
A anglefrom cylindrical coordinates0 EOE 2T
Remark If Cr 0,7 is the cylindrical coordinates of E
r psuitthen
goodIn particular z2 r2 p2
Then
iyX rest paisfosoy raise f singserifz z piesof
rectangular spherical
cylindrical
Remardi If c is aconstant then
g c c o describes a sphereofradio c describes a halfplane vertical
of C describes
we 7 axis if 0
if ITI u mi
xy planecone otherwise
u
Volueneelement
ydV dyndydt rdrdodz.tvsink
e 4 is thepolarof 1Fr
psuicfdpdodoyaroo pau.IOGo pauifdo
Pool palo
tag ofeg i Convertthe following into sphericalcoordinates
s Hy't Cz 15 1 sphere
C f Eye Cone
Sole l sub x pstoolGOy psing si 0z pw
into x y't Cz 15 1
fsiigceoo tpsu.iosin2O Cpceso lJ 7
p2sing t p654 2pm0 1 1
p spoof O
ie f p zoo o togetherwith Peoof zca f o is apointonly
c sub the formula into 7 5 27
gasof greig sie xi yt F pki.to970 050 Arioso
For pto ie not the origins6s silo tanto l
a 5
I XX
eg seeegos
Volume of ice creamcare againin spherical coordinates
0
Sod The ice cream cone I is given byO E f ETOE OfE Is don'tmissthisOef ENT
mvolt f fo 5sinoldpdoldo
mucheasierthan cylindrical do Sottsingdol Steady Is check
ease fix y yf f if exists 9 03
0 If CXy D 90,0
f is contusions and infact wedon'tneed this as value of
f at one point doesn't affectthe du
let D unit ball centered at aigui intersecting with theO loist octant
µ EThen D can berepresentedinspherical coordinates N
OE p s yx o F E
o E toEEo so 5
siu95_psiu4Ahdfcxiyzj_x45Cx.yto y E Tasf 20 f o s f is continuous and the favela iscorrect even for Cay 73 0
E i ftHence fixy dV sing fainpdpd4do
O o O Te dV
Izo
siu3od SPdp
2check
If we want to calculate the average off over D
we need to calculate Vol D too
In our case VoeCD f Vol unitsphere t IIHence
average off overD SIfCxisz2dVt
age Improper integrals
let fairy f yzz IT9 9 µ p
mbmtd
over unit ball B Kp4 A OffEl's
d Does lesion this Adv exist
where Be ftp 0 oJ oEpEE
dis Does fujio qescxis.edu exist
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