math 101
Semester 2
January
2024
, Week 1 -
definite integral
Sf(x) ax ↑ Area between the curve & the X-axis from (a , b)
Y
- a 7 * b
~
L
X
taking
and
2 ->
small
adding
summation
sections
up,
& defining functions using
?
definite integrals
dX
TAKE THE INTEGRAL
TO FIND FUNCTION
Ex Sketch &
simplify Alx) =
So tat x0
I
[xdx
Y L bounds
unt >
- A(X) =
tasheatt
=
2x2
AX =
Pa t
O *
Approximating Integrals
①
Right Riemann Sum >
Definite Riemann Integral one definition of definite integral is
-
1 f(x)ax =
Ax(fx ,
+ fX2 + ..
) fX4
&
as limit of Riemann Sum
+ (xi) ·
A
-
x4
f
x
Cf(xdx = m x
-x
!
O
-
1 X
Summation Notation Geometric sum
-
↳
EX . 4 + 6 + 8 + 10 + 12 OR 2 + 2 + 2 + 2 + 2 S =
1 + p + r2 +... + un
S =
rn -
1 ↓ ↓ ↳
Si
5
r
-
1
& S =
:
can
① ② ③
add constant factor constant commutative addition
S S S
"Sc = 10
= G(i
i 1
=
+ 2)
= 2
i= 1
, Week -
2
Integration Using Symmetry
Even Enx Odd En X
f(x) =
f) -
X) Symmetric f(x) = -
f(X) symmetrical
even around y-axis around origin
S
f(x) f) X) ↓
↳
= -
·
↳
Sef(x(ax =
2(of(x)dX
(af(x)ax =
2(f(x)dx Sf(xax = 0
odd Basic Integral Properties
f(x)
=
-
f(x) 0Sakf(x)dx = kSaf(x)aX
↳ ② S (f(x) +
g(x))dx
=
SaPf(X(ax +
*
Sa g(x)ax
Saf(x(aX =
O ②
Sf(x)dX =
- Sif(x)dX
①
Saf(x) =
Saf(x)dX + Sif(x1dX
Fundamental Theorem of Calculus I apDlY elmitS
Saf(t)d + A'(X) 11mD h) A(x)
=
A(x) = A(x +
-
n= 0 n
curve approcess em f(x)
=
=
FTC 1
A(x) =
Saf(t)dt
Y
a xynth
me13 A(X) =
Saf(t)dt where est) is continuous
A'(x) =
f(x) ↓
A'(X) =
f(x)
Fundamental Theorem of Calculus #
Assume f cont on La ,
bJ , let F(X) be fux statement
F(X) =
f(x) (a , b) (such that F's called antiderivative of f)
FTC 2 Then for a = X = b
Integrals & derivatives
Saflt)at F(X) F(a)
23 Sf(t)dt F(a)
= -
= F(x) -
*
inverses of each other
-
All off
Indefinite Integrals possible antiderivatives
* NO BOUNDARIESA
↑ (x) is fax statement
> (f(x)ax =
F(x) +
2
↑
F'(x) =
f(x) then
abitrary constant
using linearity of integrals
↓
P(X) =
anxn +
an - ,X + ...
+
ax + do ant0
-
Sp(x) =
an
Semester 2
January
2024
, Week 1 -
definite integral
Sf(x) ax ↑ Area between the curve & the X-axis from (a , b)
Y
- a 7 * b
~
L
X
taking
and
2 ->
small
adding
summation
sections
up,
& defining functions using
?
definite integrals
dX
TAKE THE INTEGRAL
TO FIND FUNCTION
Ex Sketch &
simplify Alx) =
So tat x0
I
[xdx
Y L bounds
unt >
- A(X) =
tasheatt
=
2x2
AX =
Pa t
O *
Approximating Integrals
①
Right Riemann Sum >
Definite Riemann Integral one definition of definite integral is
-
1 f(x)ax =
Ax(fx ,
+ fX2 + ..
) fX4
&
as limit of Riemann Sum
+ (xi) ·
A
-
x4
f
x
Cf(xdx = m x
-x
!
O
-
1 X
Summation Notation Geometric sum
-
↳
EX . 4 + 6 + 8 + 10 + 12 OR 2 + 2 + 2 + 2 + 2 S =
1 + p + r2 +... + un
S =
rn -
1 ↓ ↓ ↳
Si
5
r
-
1
& S =
:
can
① ② ③
add constant factor constant commutative addition
S S S
"Sc = 10
= G(i
i 1
=
+ 2)
= 2
i= 1
, Week -
2
Integration Using Symmetry
Even Enx Odd En X
f(x) =
f) -
X) Symmetric f(x) = -
f(X) symmetrical
even around y-axis around origin
S
f(x) f) X) ↓
↳
= -
·
↳
Sef(x(ax =
2(of(x)dX
(af(x)ax =
2(f(x)dx Sf(xax = 0
odd Basic Integral Properties
f(x)
=
-
f(x) 0Sakf(x)dx = kSaf(x)aX
↳ ② S (f(x) +
g(x))dx
=
SaPf(X(ax +
*
Sa g(x)ax
Saf(x(aX =
O ②
Sf(x)dX =
- Sif(x)dX
①
Saf(x) =
Saf(x)dX + Sif(x1dX
Fundamental Theorem of Calculus I apDlY elmitS
Saf(t)d + A'(X) 11mD h) A(x)
=
A(x) = A(x +
-
n= 0 n
curve approcess em f(x)
=
=
FTC 1
A(x) =
Saf(t)dt
Y
a xynth
me13 A(X) =
Saf(t)dt where est) is continuous
A'(x) =
f(x) ↓
A'(X) =
f(x)
Fundamental Theorem of Calculus #
Assume f cont on La ,
bJ , let F(X) be fux statement
F(X) =
f(x) (a , b) (such that F's called antiderivative of f)
FTC 2 Then for a = X = b
Integrals & derivatives
Saflt)at F(X) F(a)
23 Sf(t)dt F(a)
= -
= F(x) -
*
inverses of each other
-
All off
Indefinite Integrals possible antiderivatives
* NO BOUNDARIESA
↑ (x) is fax statement
> (f(x)ax =
F(x) +
2
↑
F'(x) =
f(x) then
abitrary constant
using linearity of integrals
↓
P(X) =
anxn +
an - ,X + ...
+
ax + do ant0
-
Sp(x) =
an
