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