EI - University of Pittsburghkaveh/Lecture-MATH0230-Sep15-2017.pdf · Sep. 1512017 Comparison...
Transcript of EI - University of Pittsburghkaveh/Lecture-MATH0230-Sep15-2017.pdf · Sep. 1512017 Comparison...
Sep . 1512017
.Comparison tech .for improper integrals .
EI( # 41 6.6 )
✓
§¥f,dx
#on divergent ?
.
Recall general fact theorem :
aA
If of fcx ) £ gc × ) & af 9 '×)d× Com . ⇒ ffcxsdxC for × in our range of integral ) a Corn .
( a ,oo )
( Also if § fandx divergent ⇒ a§g(x)d× divergent . )
Iaea 'sRecall §×÷dx convergent D
( bft.gl/#xfdiengxt@n no worry because
§×3÷,
dx is just
µ××z,dx £ §txzdx Com.
aconst .
number.
e §¥,is some
const . number}⇒§×ITd× com
.
EEC# 43 6.6 )
,J°××⇐p÷y_dx corn . or
dir. ?
Xtl ×- > -
of-
#= Iz = xt
- X
( XZI ) a
xtl
|¥d× is divergent ⇒
,
) -×y_×dxdivergent
÷ap= Applications of integrations .
( done with integration techniques D)
•Reduce computation of different quantitiesto computation of a ( definite ) integral .
- Area between curves.
- Volumes ( obtained by notation of a curve ) .
- Quantities from physics like w°¥ .
Genenalfautltheoremfcx) z gcx ) for
afxsb.gs#tYkg?fIeafdegtTnettEensII?&X=bis:
-
§ Has-gcxDd×
Note : When applying this make sure fzg .
E×#7t)
y=5וx2Area of region
bounded by :y=×
4- . - - . . •¥114MIE.int#ions :¥€4x=x2⇒ II0g
Sx -x2 z × when o 1×5-4
4Answer : g ( sxii ) - × dx
o
( In some questions , they may ask to just write
the correct integral ,without computing it )
E± Find area enclosed by the graphs
of Y = sin ( × ) & y =-3 Sin (F)& lines X=O & X = 317 .
Tit, get a rough idea how the graphs look like
.
Athayde.IS?IIxIsn"⇒ .
( to compute the int . / compute §"
sin # dx &
§"3sin(g) dx separately .)
.
Ex.
( Example 4 in 7. 1)
Area enclosed by y=×- I & YZ 2×+6 .
yet1
III.
ideat⇒×→%¢#¥¥-3 = y+ 1 ¥E- y -4=0
/7=4 and -2
( × = 5 and -1 )
WRIT function of y ( sowgixtdfyrde)4)(7+1) - ( ¥-3 ) dy
y= - 2
ingraining→€#¥n#,I"*t.fi#more work
.
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