Wed 111n this section we'll learn hw to compute various volumes of solids usingintegrals Then we'll apply these techniques to several architectural structures

Areabetweercumeski.li I.Eft

abfflxs glxDdx fab fan dx fabgal DXThis ever works if the region is below or straddles the x axis

Examplet Find the area between the curves of Hx 5 X d g1 22 3

First find points of intersection 3

5 x X 3

8 2 2 x 12

Area Iif xD hi 3 d s x

Iid 2 4dx 8 3 31 46 E f k Es u IzExample Find the area between the curves y TX y X 2 and

the x axis

D r i y x 2Method Integrate w r 1 xy rx

nothotht.ggudtohe hth to NHt

2 1,4

i

firx o dx fix dx x rx3I E1 rx k 2 dx L x 2 dx Xk Eat2x

Ff Et 8 Eri 2

8 1848 Frs 2 4 tf Ers 12 E_ZTotal area DMethod Integrate w.it y z y f

4 5

FArea fo ly 2 y dy

I 12g F 4

y x 2 X y 12

2 4 E 0 0 OD

much easier

Volumes by slicing

well now learn to derive classic formulas for volumes such asVol cone IT r h and vol sphere ar

The method is in some sense a 3D version of how Archimedes computed

the area b cite

OD.DEJ

Voluneofa Idea Slice the cone into layers like a

R

gwedding cake Each layer is a cylinder

Ry

1h 7 y

ft 4 Eyr r

_dyArea a r h radius r X valve EhyTiffg dy T y

iii ivg.si royal s ErdyThis y dy TRIP 1 TIME E STRICT r

dameofahemispheetz.LYTRY 3 FR Iy R

dy 9

radius r x value TRYvoi e Toth THEYdy 2 y2 R

l x tryVol of hemisphere for Vol ie

for a R y dy for R ayy dy

R'y Tig a R TRI V

Example Consider the solid formed by revolving the curve y rxraround the x axis from x O to X 4 Find its volume

i ii i f iii

Tix dx

Vol IO Vol LII Tix dx a'Elo

These are called solidsafintionThe method we've been doing is called the diskmethat

We can do other shapes

Ex

if

Exe

gg fH no

f a Td Tie dx

Wed146E Gabrin FCM ty from x l to x

ih

First we need to see what are called improperintegrals i.e integrating

over an asymptote or where a limit is x

3igiden treat as an ordinary number

Example f dx In I In x ln I x o

Technically I tx dx him J dx him Inxp Liz h b

sit dx txt to ft OH

Mjzach to Gabriel's horn

7Vol 02 L Vol O If a dx fitJust for fun we can compute thesurface area

ft

iiiI

p t

Surfacearea LardSurface area L a r d

2x E DX f 2 a r Ftfx dx

Thus surface area SAC Io

s SAF f 2TEDx

2ark x 2ar In no In 1

Thus Gabriel horn has finite volume but infinite surface area

We can fill it with paint but not paint the whole surface

The following technique is sometimes called volume by washers

EI Compute the volume of the region b w y x and y X

I

rammerTedx

vol a Mdxy x2 stay y x y

3igiden

I l l

l I

foHy Tey dy Jo Ti d fo t y dx

CI THI E fo o

Fi 11 8Another way to find the volume of the previous solid is the shellmethod

r x dxEe

I x x

aanVol Lar h dy 2 x x d

Vol Uae forollfo 2ax x Y dy tix 2 3 dx

f k 11 2K 11

2 s 4 fo o

Archeryth

Good Find the length of a curve y fix from x a to x b

ogs are 6

Id infinitesimal

dyare length

dx

Are lentissimo as figsneedformula

ds cdxtldyF fdfdy.de olxTdyYfddffF

ffdaxTtfdHld ltfkdxThus arclemgthfabltfkd.TLMon 11 11

Examples

Find the arc length of y fX3 X from x O to x

to lt x dx si f

i

ht u It Ex du Ii dx dx TduX y

ru Ia du 1 u du a IIIt Ex It 10 Izz I too i

9.073

Renate often one of these integrals fifty d ends up

being too complicated leg no closed form Solin or we havent

learned the method For there WolframAlpha is helpful

Exe Compute the arc length of one full cycle of sin

Yifan dx

Ft dx