theorem (second fundamental theorem of calculus)
TRANSCRIPT
Theorem (Second Fundamental theorem of calculus).∫ t=β
t=α
f (t)dt
1
Theorem (Second Fundamental theorem of calculus).∫ t=β
t=α
f (t)dt =
2
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t)∫ t=β
t=α
f (t)dt =
3
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ t=β
t=α
f (t)dt =
4
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ t=β
t=α
f (t)dt = F (β)− F (α)
5
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ t=β
t=α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
6
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
7
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t),
8
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt
9
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
10
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
11
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
12
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
2.∫tndt = tn+1
n+1
13
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
2.∫tndt = tn+1
n+1
3.∫cos(t)dt = sin(t)
14
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
2.∫tndt = tn+1
n+1
3.∫cos(t)dt = sin(t)
4.∫sin(t)dt = − cos(t)
15
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
2.∫tndt = tn+1
n+1
3.∫cos(t)dt = sin(t)
4.∫sin(t)dt = − cos(t)
Example. γ(t) = (r cos(t), r sin(t))
16
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
2.∫tndt = tn+1
n+1
3.∫cos(t)dt = sin(t)
4.∫sin(t)dt = − cos(t)
Example. γ(t) = (r cos(t), r sin(t))γ̇(t) = (−r sin(t), r cos(t))
17
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
2.∫tndt = tn+1
n+1
3.∫cos(t)dt = sin(t)
4.∫sin(t)dt = − cos(t)
Example. γ(t) = (r cos(t), r sin(t))γ̇(t) = (−r sin(t), r cos(t))‖γ̇(t)‖ = r
18
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
2.∫tndt = tn+1
n+1
3.∫cos(t)dt = sin(t)
4.∫sin(t)dt = − cos(t)
Example. γ(t) = (r cos(t), r sin(t))γ̇(t) = (−r sin(t), r cos(t))‖γ̇(t)‖ = r∫ π0 ‖γ̇(t)‖dt =
∫ π0 rdt = πr
19
Theorem (Second Fundamental theorem of calculus).If f (t) = F ′(t) for some F∫ β
α
f (t)dt =
∣∣∣∣t=βt=α
F (t) = F (β)− F (α)
Notation: Often “t =” is dropped
Definition. Anti-derivative of f (t), denoted,∫f (t)dt = F (t)
where F (t) is so that F ′(t) = f (t).
Examples.
1.∫tndt = tn+1
n+1
2.∫tndt = tn+1
n+1
3.∫cos(t)dt = sin(t)
4.∫sin(t)dt = − cos(t)
Example. γ(t) = (r cos(t), r sin(t))γ̇(t) = (−r sin(t), r cos(t))‖γ̇(t)‖ = r∫ π0 ‖γ̇(t)‖dt =
∫ π0 rdt = πr
20
f1(t) = F ′1(t)
21
f1(t) = F ′1(t) and f2(t) = F ′2(t)
22
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t) + f2(t)dt
23
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t) + f2(t)dt
f1(t) + f2(t)
24
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t) + f2(t)dt
f1(t) + f2(t) = F ′1(t) + F ′2(t)
25
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t) + f2(t)dt
f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))′
26
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t) + f2(t)dt = F1(t) + F2(t)
f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))′
27
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
28
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
29
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt
30
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t)
31
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
32
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t)
33
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)− F2(t)
34
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
35
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt
36
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)
37
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
38
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!
39
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′
40
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)
41
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′
42
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t)+
43
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
44
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t)
45
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t)+
46
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)
47
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t)
48
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−
49
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
50
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
Example.∫t cos(t)
51
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
Example.∫t cos(t) =
52
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
Example.∫t︸︷︷︸f(t)
cos(t) =
53
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
Example.∫t︸︷︷︸f(t)
cos(t)︸ ︷︷ ︸g′(t)
=
54
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
Example.∫t︸︷︷︸f(t)
cos(t)︸ ︷︷ ︸g′(t)
=
where g(t) = sin(t)
55
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
Example.∫t︸︷︷︸f(t)
cos(t)︸ ︷︷ ︸g′(t)
= t sin(t)
where g(t) = sin(t)
56
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
Example.∫t︸︷︷︸f(t)
cos(t)︸ ︷︷ ︸g′(t)
= t sin(t)−∫
sin(t)
where g(t) = sin(t)
57
f1(t) = F ′1(t) and f2(t) = F ′2(t)∫f1(t)+f2(t)dt = F1(t)+F2(t) =
∫f1(t)+
∫f2(t)
because,f1(t) + f2(t) = F ′1(t) + F ′2(t) = (F1(t) + F2(t))
′
Example.∫cos(t) + t3dt = sin(t) +
t4
4
Similarly,∫f1(t)− f2(t) = F1(t)−F2(t) =
∫f1(t)−
∫f2(t)
Example.∫cos(t)− t3dt = sin(t)− t4
4
Products need care!(f (t)g(t))′ = f ′(t)g(t) + f (t)g′(t)∫
(f (t)g(t))′ =
∫f ′(t)g(t) +
∫f (t)g′(t)
f (t)g(t) =
∫f ′(t)g(t) +
∫f (t)g′(t)∫
f (t)g′(t) = f (t)g(t)−∫f ′(t)g(t)
Example.∫t︸︷︷︸f(t)
cos(t)︸ ︷︷ ︸g′(t)
= t sin(t)−∫
sin(t) = t sin(t)+ cos(t)
where g(t) = sin(t)
58
Substitution rule
59
Substitution rule
s
60
Substitution rule
s : [α, β]
61
Substitution rule
s : [α, β]→ R
62
Substitution rule
s : [α, β]→ Rφ
63
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]
64
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
65
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
(s(φ(t̃)))
66
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
(s(φ(t̃)))′
67
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
(s(φ(t̃)))′ = s′(φ(t̃))
68
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
(s(φ(t̃)))′ = s′(φ(t̃))φ′(t̃)
69
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′
70
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃))φ′(t̃)
71
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃))φ′(t̃)dt̃
72
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃))φ′(t̃)dt̃ =
∫ β̃α̃ (s(φ(t̃)))
′
73
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃))φ′(t̃)dt̃ =
∫ β̃α̃ (s(φ(t̃)))
′dt̃
74
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃))φ′(t̃)dt̃ =
∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃))
75
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃))φ′(t̃)dt̃ =
∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ φ(β̃)φ(α̃)
76
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃))φ′(t̃)dt̃ =
∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ φ(β̃)φ(α̃) s
′(t)
77
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃))φ′(t̃)dt̃ =
∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ φ(β̃)φ(α̃) s
′(t)dt
78
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃ =∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ φ(β̃)φ(α̃) s
′(t)dt
79
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ β̃α̃ s′(φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ φ(β̃)φ(α̃) s
′(t)dt
80
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ s
′(φ(t̃)︸︷︷︸t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ φ(β̃)φ(α̃) s
′(t)dt
81
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ s
′(φ(t̃)︸︷︷︸t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ t=φ(β̃)t=φ(α̃) s
′(t)dt
82
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ s
′(φ(t̃)︸︷︷︸t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ t=φ(β̃)t=φ(α̃) s
′(t)dt
Informally:Substituting, t = φ(t̃)
83
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ s
′(φ(t̃)︸︷︷︸t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ t=φ(β̃)t=φ(α̃) s
′(t)dt
Informally:Substituting, t = φ(t̃)dtdt̃= φ′(t̃)
84
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ s
′(φ(t̃)︸︷︷︸t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ t=φ(β̃)t=φ(α̃) s
′(t)dt
Informally:Substituting, t = φ(t̃)dtdt̃= φ′(t̃)
dt = φ′(t̃)dt̃
85
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ s
′(φ(t̃)︸︷︷︸t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ β̃α̃ (s(φ(t̃)))
′dt̃ = s(φ(β̃))− s(φ(α̃)) =∫ t=φ(β̃)t=φ(α̃) s
′(t)dt
Informally:Substituting, t = φ(t̃)dtdt̃= φ′(t̃)
dt = φ′(t̃)dt̃
86
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ s
′(φ(t̃)︸︷︷︸t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) s
′(t)dt
Informally:Substituting, t = φ(t̃)dtdt̃= φ′(t̃)
dt = φ′(t̃)dt̃
87
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
Informally:Substituting, t = φ(t̃)dtdt̃= φ′(t̃)
dt = φ′(t̃)dt̃
88
Substitution rule
s : [α, β]→ Rφ : [α̃, β̃]→ [α, β]
s′(φ(t̃))φ′(t̃) = (s(φ(t̃)))′∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
Informally:Substituting, t = φ(t̃)dtdt̃= φ′(t̃)
dt = φ′(t̃)dt̃
89
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))
90
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃)
91
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)
92
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖
93
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
94
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0
95
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0‖ ˙̃γ(t̃)‖
96
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))‖φ′(t̃)
97
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))‖φ′(t̃)
98
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃))‖φ
′(t̃)dt̃
99
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃
100
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃︸ ︷︷ ︸dt
101
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ φ(β̃)φ(α̃)
102
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ φ(β̃)φ(α̃) ‖γ̇(t)‖
103
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ φ(β̃)φ(α̃) ‖γ̇(t)‖dt
104
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ φ(β̃)φ(α̃) ‖γ̇(t)‖dt
We have proved,
105
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ φ(β̃)φ(α̃) ‖γ̇(t)‖dt
We have proved,
Theorem. The arc length
106
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ φ(β̃)φ(α̃) ‖γ̇(t)‖dt
We have proved,
Theorem. The arc length is invariant
107
∫ t̃=β̃t̃=α̃ F (φ(t̃)︸︷︷︸
t
)φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ t=φ(β̃)t=φ(α̃) F (t)dt
γ̃(t̃) = γ(φ(t̃))˙̃γ(t̃) = γ̇(φ(t̃))φ′(t̃)‖ ˙̃γ(t̃)‖ = ‖γ̇(φ(t̃))φ′(t̃)‖ = ‖γ̇(φ(t̃))‖|φ′(t̃)|
Assume, φ′(t) > 0∫ β̃α̃ ‖ ˙̃γ(t̃)‖dt̃ =
∫ β̃α̃ ‖γ̇(φ(t̃)︸︷︷︸
t
)‖φ′(t̃)dt̃︸ ︷︷ ︸dt
=∫ φ(β̃)φ(α̃) ‖γ̇(t)‖dt
We have proved,
Theorem. The arc length is invariant underreparametrization.
108