4.8 improper integrals - stem math & calculus...this was created by keenan xavier lee, 2013. see...
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This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
4.8 Improper Integrals
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
010 Area - Integration
Let's consider the function : fate'×
,Graph the function and find the
area between XO and x=
:I6
5
:
I#*###'
-8 -7 - 6 ' 5 -4 .2 ' ' ifI
Need to integrate :
§e' '' dx = . } e
" du =.eu}=E×}= Ee' ' "
) - Ee of =
¥+1 .
U = - ×
dui -1 du- du = Idu
We were able to use integration to be able to find the area between
2 boundaries.
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
ned Improper Integrals
Let's consider the past graph of the function :
4 i Befort finding area between 2
aboundaries
iCowigfindingarea
when one
sided is unbounded or¥*¥¥*he¥¥¥;#⇒both sides are unbounded
.:#Infinite Integrals ,
whereanintnityiseitheratoneorbothlinitsofintegration@lfflDiscontinnonsoncyoh.t
hen §fKd×=lim •§fK)dx
aa
Otfflxiscontinuous onto,D,
then ffkldxilim fflxjdxt.at
�3�lfftxiscontinnonsontaohthen;fcx
)dx=§fHd×t§fHd×lwherecissome
- a
real number)
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
Solutions can be either :
@ convergent- the limitexist I is a real number
.
to divergent - the limit fails to exist I is infinite to or a ).
Suppose we want to find the area starting from 0 going towards positive
infinity (a) .
How do we find the area ?
t
§e d× =
tiny.
D e '×dx Him fteudusfnwouft= fig.
- etftooA- -x
du = -1 dx- du = dx
=
find.[ - et .
-
e°] = fin,a@wy1 = 0+1 = 1,
convergent
[Examples] Integrate.
�1� I¥= final team tent 's dating. ¥ It
two ÷⇒tnt±nE÷-
i¥=¥⇒#att¥= Ot £ = ±z
,
convergent .
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
�3� }.EE#=titma&c*3Ex+TdxHx?E=xtI+t3a2=A(xt1)tB(xt3
)2 = AxtAtBxt3B
antxait :L:'I¥B⇒ . LIFE'33.
}÷}zBFMDA : Oiatk )
- 1=At '
time ! ,÷3tx÷±dx=fm
.tn#3IHnKttDote=fmjtnH.iHHntttHtElnH&tnkD=fing.lnltttttsttn3=fmwlnltItHTo0ttn3=lnHthnl3I=H1
,convergent .
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
0IedxtotInafexdxQienytexdxerduatea0diewfexdx.fme.ex5tefimwe-etItietatet-1-OT1eraluate@lgeafexdxytmjaex3otetfmj.et
- i =thing
.
et . 2
= a - 1 = N
So,
1 to = a; divergent .