TT Nov2 - Mathematical Sciencesmacaule/classes/f18_math1060/f18... · 2018. 11. 13. ·...
Transcript of TT Nov2 - Mathematical Sciencesmacaule/classes/f18_math1060/f18... · 2018. 11. 13. ·...
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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
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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
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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
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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
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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
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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