Rew
·
Definite Integral -
signed area under a curve
↳ defined using Riemann sums
·
Properties of Definite Integrals
1
. If f is cont oh Cabb, then f(x) Ex exists
/faxdx -So f(x)bx
.
4
. If f and
2 .If bla =
g are integrable on [abs , then: .
Sb(f(x) g(x)) (a f(x)(x + (a g(x)6x
+ =
. If
5 a <bl
,
J fixdx = fixx 50
+
X
. fixdx
3 = ca f(x)x
·
Antidenrative -
F is the anti-derivative of f
·
F'(x) =
f(x) on Casb]
·
Indefinite Integral -
(f(x)(x = F(x) + C
Eis()(x (( * * ((x = = ((x -
4x+(x =
(x(x -
4(2x
=+
· FTCs
1 .
Ex flett f(x) =
for all x E Cab]
. f(x(x F(b)
2 = -
F(a) = F(x))"a
, Sect 5 6 .
-
Net Change
Greatapplication
FI be function on C-a , 0)
↳
> s' is a cont function
Net change of sct) over an interval (t te)
,
is S(t)-S(t)
·
Theorem -
Net change of sct) over CE . E] is sitSt
·
basically : = SEE
s(t)-s(t)
-T
*
untider of s
-
s is an
FTC : S'(t)ft =
SCE)-S(t ) ,
·
Examples : Sis usually :
ovolume of water n a tank displacement of south opopulation of south
·
Initial Value Problems :
#1 ) .
Say y is a function of
E, satisfying flts = t + sint ,
y10) =
.
U
Method
Find +(t) . (ify is displ, y is velo cruc of displ)
+
: Ftc : + (t) -
+ (0) =
% Y'(3363
method
·
#2
:
+ is an antider of y .
So to find y ,
we need to .
find antifer tor sint
#
y(t) ((t2 + sit)0t =
cost C displ= Velo
-
+ *
=
Voc of
-
row if dist .
=
Speed
now... y(0) = 0 -
cos O +
speej = 1 vctl
0 =
- 1 + 2 so ...
antider of displ
velo
.
c = 1 so ... y(t) = E -
cost + 1 antifer of
speed
= Gist .
1 .
((t)
#2 ) A particle has
. velo.(H) = (t- YE + 3t) mis .
Find displ .
of particle
17
↳ Over [0. 3] and "Gist. travelled by the particle over [03].
displ .: S(3)-3(0) Jo's'
= LESt =
ty 4t 3
- +
= VISE - meters =
It
signed
area is the integral &*
of velo which is the displ ,
displ over 20, 33 : <(3)-S10) = bet
-j(t2 ye +3t)8t
h
.
which s the net change = change
t
-
in positive position
, Gist. Traveled == GV-
=
do'viesot -" uctiJt
=
So Ct3 Ye 3t)87 -"(t3
- + - -
472+ 37)6t
=>
Ejy 4ty
, +
3ty)) (ty 45y 3ty2) )
-
-
+
,
dist= 3712 m
t
#3) Rate
.
of Flow of water into a tank is r(t) =
be eal/hr
.
-
(1) If the tank initially contained I gal of water, what is the volume of water in the
tank after 4 hrs.
t= w
(11) If the water is poured for eternity
-
,
how much water is in the tank at the end
of eternity"
ii.)
change
in
volume
= So Crow of vo rit) =
get
=
350e to
=
3(- e- t)) !
= +
3 gal .
so ... total vol =
2 +3 = 5 ga).
·
Definite Integral -
signed area under a curve
↳ defined using Riemann sums
·
Properties of Definite Integrals
1
. If f is cont oh Cabb, then f(x) Ex exists
/faxdx -So f(x)bx
.
4
. If f and
2 .If bla =
g are integrable on [abs , then: .
Sb(f(x) g(x)) (a f(x)(x + (a g(x)6x
+ =
. If
5 a <bl
,
J fixdx = fixx 50
+
X
. fixdx
3 = ca f(x)x
·
Antidenrative -
F is the anti-derivative of f
·
F'(x) =
f(x) on Casb]
·
Indefinite Integral -
(f(x)(x = F(x) + C
Eis()(x (( * * ((x = = ((x -
4x+(x =
(x(x -
4(2x
=+
· FTCs
1 .
Ex flett f(x) =
for all x E Cab]
. f(x(x F(b)
2 = -
F(a) = F(x))"a
, Sect 5 6 .
-
Net Change
Greatapplication
FI be function on C-a , 0)
↳
> s' is a cont function
Net change of sct) over an interval (t te)
,
is S(t)-S(t)
·
Theorem -
Net change of sct) over CE . E] is sitSt
·
basically : = SEE
s(t)-s(t)
-T
*
untider of s
-
s is an
FTC : S'(t)ft =
SCE)-S(t ) ,
·
Examples : Sis usually :
ovolume of water n a tank displacement of south opopulation of south
·
Initial Value Problems :
#1 ) .
Say y is a function of
E, satisfying flts = t + sint ,
y10) =
.
U
Method
Find +(t) . (ify is displ, y is velo cruc of displ)
+
: Ftc : + (t) -
+ (0) =
% Y'(3363
method
·
#2
:
+ is an antider of y .
So to find y ,
we need to .
find antifer tor sint
#
y(t) ((t2 + sit)0t =
cost C displ= Velo
-
+ *
=
Voc of
-
row if dist .
=
Speed
now... y(0) = 0 -
cos O +
speej = 1 vctl
0 =
- 1 + 2 so ...
antider of displ
velo
.
c = 1 so ... y(t) = E -
cost + 1 antifer of
speed
= Gist .
1 .
((t)
#2 ) A particle has
. velo.(H) = (t- YE + 3t) mis .
Find displ .
of particle
17
↳ Over [0. 3] and "Gist. travelled by the particle over [03].
displ .: S(3)-3(0) Jo's'
= LESt =
ty 4t 3
- +
= VISE - meters =
It
signed
area is the integral &*
of velo which is the displ ,
displ over 20, 33 : <(3)-S10) = bet
-j(t2 ye +3t)8t
h
.
which s the net change = change
t
-
in positive position
, Gist. Traveled == GV-
=
do'viesot -" uctiJt
=
So Ct3 Ye 3t)87 -"(t3
- + - -
472+ 37)6t
=>
Ejy 4ty
, +
3ty)) (ty 45y 3ty2) )
-
-
+
,
dist= 3712 m
t
#3) Rate
.
of Flow of water into a tank is r(t) =
be eal/hr
.
-
(1) If the tank initially contained I gal of water, what is the volume of water in the
tank after 4 hrs.
t= w
(11) If the water is poured for eternity
-
,
how much water is in the tank at the end
of eternity"
ii.)
change
in
volume
= So Crow of vo rit) =
get
=
350e to
=
3(- e- t)) !
= +
3 gal .
so ... total vol =
2 +3 = 5 ga).
