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Overzicht Stof Deel 2 (samenvatting) Analyse 1 NA
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Calculus 2 Notes (Integral Calculus)
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Summary Calculus 2 (2WBB0) 2020/2021
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Complete summary Calculus 2
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Oneigenlijke integralen:Improper integrals of type 1:
If f continuous on [a, ∞), we define the improper integral of f over [a, ∞) as a limit of proper integrals
SEHdx-hi.%a7sdx.bjtlsdx-nli.s/b5HdxSbaEl dx-
If f continuous on (-∞, b], then we define
==> als de limiet bestaat, dan convergeert de oneigenlijke integraal; als de limiet niet bestaat, dan divergeert de
oneigenlijke integraal
Improper integrals of type 2:
if f continuous on interval (a,b] and is possibly unbounded near a, we define the improper integral:
↳ dx
Linga ,
if f continuous on [a,b) and is possibly unbounded near b, we define
J !EHdx= Ling f !FHdx -
==> als de limiet bestaat, dan convergeert de oneigenlijke integraal; als de limiet niet bestaat, dan
divergeert de oneigenlijke integraal
p-integrals:
if 0 < a < ∞, then
IT
Ja {
converges to
?
,
is p
.
>
Ca) -
pdx
diverges to no
if p El
pa 'T if pe
{
,
converses to
(b) J! ×
-
pa ,
diverges to re if P ? I
, Zo ook,
↳ Pdx converge ert voor
p > -
l J Pdx diver geert voor alle p
J Pdx Carver geert voor p , -
,
J -
Pdx diver
geert voor alle p
,
A comparison theorem for integrals
Let -∞ < a < b < ∞, and suppose that functions f and g continuous on interval (a,b) and satisfy 0 < f(x) < g(x).
if
Jbagcxsdx converges, then so does fbufcxsdx
and
Sbajcxsdxsfbagcxdx
Equivalently, if J!gCx)dx diverges to ∞, then so does Jbaflxsdx ( and satisfy gas >
5-
Cx) > o )
Vb .
Gu na voor
Xu
Welke a e IR
f! d-
convergent .
De integrand Inc ,
is continue
op ( o te) ,
.
In (E) = -
I
,
dues In ( x) E -
I en x
"
> o .
Xu
Stel
fcxj-eh@iudutfkS2oophetdomeinenglxS-xa.Dun
"
genet : O c E X
y ,
×
Jxhdx convergent J J
xu
We we ten dat
By comparison voor a > -
t .
converge ert
-
Inch
dx = -
,
dx dan 00k
Nu toner we aan dat de integrant diver geert voor a -_ -
l :
In 11h (E) I
J !↳d× ft In
'
lnllncxsl
I! ! lnlultc dx=
tiny
=
-
-
du lnllucxjltc
-
→ → = s
-
-
× , a.
i.
Type 2
Xh
J
-
'
x
f
dx diver geert by comparison
Oman .
, diverge ren alle dx voor al -
look want > > 0
Ink) Ink)
,
Dus de integrant convergeert voor a > -
I en diver geert voor as
-
I
vb 2 .
Depaul of de oneiyenhjke integrant J! ! × + ×
converge ert of diver geert .
De integrant is
"
improper of both types
"
,
dies we Kuennen Schryver
J! ! × + ×
t
)! ! × + ×
Op (o ,
I ] is xstx > X
,
dins :
floe !
"
{ ou !
)
L →
By comparison convergent
× + ×
→ p -
- I < I →
convergent
h '
op a. → is # + × > ×
'
,
aus :
aus de
integral convergent
[ ×
! + ×
CJ s , →
→
By
E
comparison convergent
p - > I →
convergent
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