tt nov2 - mathematical sciencesmacaule/classes/f18_math1060/f18... · 2018. 11. 13. ·...
TRANSCRIPT
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