weekofoc 29 Nov2TTArenbetweercur

ffy fix fly

4 1 it

fafffxI gCxDdx affxldx fabglxldx

This ever works if the region is below or straddlesthe x axis

E Find the area between the curves of ftp 5 xardgly K 3x's

Area f F XY Ix 3 dx 5 x

f I 8 2 2 oh 8 331 46 II f 16 E 32 E

Example Find the area between the curves y Tx y x 2

and the x axis y x2

Area fi o dx f rx x dxi

X.gg

y rx

Note that we'll need to break this into 1two integrals

Then fifty o dx f rx Ix 2 dx3 x to x II 2 11

Efg T 8181 Frs 2 4 If E IzWeekofNov5 9

Aretherme_thodilnteyratew.r.t.y 2

Ann fifty yr dy y y rx

4 x

y X 22y F I It

x yt22 4 Ez Oto o I I much easier

Volumeby slicing R yh

Motivatingexampt volume of a cone i ay

I am ar HE'T

IhI I dy The radius at height y isthe

Volume of slice tr iffy2dghe 5 thx R

Volume of cylinder vd of slice height

John Fy dy foh dy Eff 1 uRjI TR

Example Volume of a hemisphere x YER

E5

dy liIVolTr2dy R yDdy 1

Vol of cylinder for vol of slice at height y

for a R g dy for uR ay dy

R'y T.IM oR uR3 j aR3

volume of other solidsofitiony r

E Consider the solid formed by revolving

tainision iii is.si ii ii to'iIiiitiiIIiiixx

voi so xxdx Elia Pdxa XDX

Sometimes this method is called the diskmethod

we can do other shapes Iumpet

jGabrielsen Tn

f y L from x L to x

iH

First we need to see what are called improper integrals ie integrating

over an asymptote or where a limit is x

Big idea treat x as any ordinary number

Example J I d In x In re ln I Aarea I

f dx txt f ft OH I 4Back to Gabriel's horn

Volume f I dy I dx a f ta dx IT

We can also compute the surface

Technically we don't know how to do this but we can show it's infinite

II

rationfifty

ZIx

f xThus the surface area is bigger than

I 2T rdx 2 dxT

f dx x

So Gabriel's horn has i e but infinitesurface area

The following method is sometimes called volumes by washers

Example Consider the region formed by rotating theD T

area between the curves y X and y x from I Y

o to I around the yaxis i IIKey idea y

x D

foTify y dy fo try dy fo t y dy

a Eto a 1 Iz EeekofNovl2 6

we can do the previous problem using a different method called shells

iii di x x

µ2 r 2

x Vol area height depth

2 tix x xDdx

Vol f rot of shellof radius x

fo 2 tix x x dx fo 2 1 2 xDdx 2x fo 2af's 4