The fundamental theorem shows that differentiation and integration are inverse processes.

1. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
.
Notes
Sec on 5.4
The Fundamental Theorem of
Calculus
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
May 2, 2011
.
.
Notes
Announcements
Today: 5.4
Wednesday 5/4: 5.5
Monday 5/9: Review and
Movie Day!
Thursday 5/12: Final
Exam, 2:00–3:50pm
.
.
Notes
Objectives
State and explain the
Fundemental Theorems
of Calculus
Use the first fundamental
theorem of calculus to
find deriva ves of
func ons defined as
integrals.
Compute the average
value of an integrable
func on over a closed
interval.
.
.
. 1
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2. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Outline
Recall: The Evalua on Theorem a/k/a 2nd FTC
The First Fundamental Theorem of Calculus
Area as a Func on
Statement and proof of 1FTC
Biographies
Differen a on of func ons defined by integrals
“Contrived” examples
Erf
Other applica ons
.
.
Notes
The definite integral as a limit
Defini on
If f is a func on defined on [a, b], the definite integral of f from a to
b is the number
∫ b ∑n
f(x) dx = lim f(ci ) ∆x
a n→∞
i=1
b−a
where ∆x = , and for each i, xi = a + i∆x, and ci is a point in
n
[xi−1 , xi ].
.
.
Notes
The definite integral as a limit
Theorem
If f is con nuous on [a, b] or if f has only finitely many jump
discon nui es, then f is integrable on [a, b]; that is, the definite
∫ b
integral f(x) dx exists and is the same for any choice of ci .
a
.
.
. 2
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3. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Big time Theorem
Theorem (The Second Fundamental Theorem of Calculus)
Suppose f is integrable on [a, b] and f = F′ for another func on F,
then ∫ b
f(x) dx = F(b) − F(a).
a
.
.
Notes
The Integral as Net Change
Corollary
If v(t) represents the velocity of a par cle moving rec linearly, then
∫ t1
v(t) dt = s(t1 ) − s(t0 ).
t0
.
.
Notes
The Integral as Net Change
Corollary
If MC(x) represents the marginal cost of making x units of a product,
then ∫ x
C(x) = C(0) + MC(q) dq.
0
.
.
. 3
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4. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
The Integral as Net Change
Corollary
If ρ(x) represents the density of a thin rod at a distance of x from its
end, then the mass of the rod up to x is
∫ x
m(x) = ρ(s) ds.
0
.
.
Notes
My first table of integrals
.
∫ ∫ ∫
[f(x) + g(x)] dx = f(x) dx + g(x) dx
∫ ∫ ∫
xn+1
xn dx = + C (n ̸= −1) cf(x) dx = c f(x) dx
∫ n+1 ∫
1
ex dx = ex + C dx = ln |x| + C
∫ ∫ x
ax
sin x dx = − cos x + C ax dx = +C
∫ ln a
∫
cos x dx = sin x + C csc2 x dx = − cot x + C
∫ ∫
sec2 x dx = tan x + C csc x cot x dx = − csc x + C
∫ ∫
1
sec x tan x dx = sec x + C √ dx = arcsin x + C
∫ 1 − x2
1
dx = arctan x + C
1 + x2
.
.
Notes
Outline
Recall: The Evalua on Theorem a/k/a 2nd FTC
The First Fundamental Theorem of Calculus
Area as a Func on
Statement and proof of 1FTC
Biographies
Differen a on of func ons defined by integrals
“Contrived” examples
Erf
Other applica ons
.
.
. 4
.

5. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Area as a Function
Example
∫ x
Let f(t) = t3 and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solu on
x
Dividing the interval [0, x] into n pieces gives ∆t = and
n
ix
ti = 0 + i∆t = .
n
x x3 x (2x)3 x (nx)3
. Rn = · 3 + · 3 + ··· + · 3
x n n n n n n
0 x4 ( 3 ) x4 [ ]2
. = 4 1 + 2 + 3 + · · · + n = 4 1 n(n + 1)
3 3 3
n n 2
.
Notes
Area as a Function
Example
∫ x
Let f(t) = t3 and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solu on
x
Dividing the interval [0, x] into n pieces gives ∆t = and
n
ix
ti = 0 + i∆t = .
n
x4 n2 (n + 1)2
. Rn =
x 4n4
0
. x4
So g(x) = lim Rn = and g′ (x) = x3 .
x→∞ 4 .
Notes
The area function in general
Let f be a func on which is integrable (i.e., con nuous or with
finitely many jump discon nui es) on [a, b]. Define
∫ x
g(x) = f(t) dt.
a
The variable is x; t is a “dummy” variable that’s integrated over.
Picture changing x and taking more of less of the region under
the curve.
Ques on: What does f tell you about g?
.
.
. 5
.

6. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Envisioning the area function
Example
y
Suppose f(t) is the func on
graphed to the right. Let
∫ x
g(x) = f(t) dt. What can
0
you say about g? .
x
2 4 6 8 10f
.
.
Notes
Features of g from f
y Interval sign monotonicity monotonicity concavity
of f of g of f of g
g [0, 2] + ↗ ↗ ⌣
.
fx
2 4 6 810 [2, 4.5] + ↗ ↘ ⌢
[4.5, 6] − ↘ ↘ ⌢
[6, 8] − ↘ ↗ ⌣
[8, 10] − ↘ → none
. We see that g is behaving a lot like an an deriva ve of f.
.
Notes
Another Big Time Theorem
Theorem (The First Fundamental Theorem of Calculus)
Let f be an integrable func on on [a, b] and define
∫ x
g(x) = f(t) dt.
a
If f is con nuous at x in (a, b), then g is differen able at x and
g′ (x) = f(x).
.
.
. 6
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7. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
∫
g(x + h) − g(x) 1 x+h
= f(t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the
minimum value of f on [x, x + h]. From §5.2 we have
∫ x+h
mh · h ≤ f(t) dt ≤ Mh · h
x
.
g(x + h) − g(x)
=⇒ mh ≤ ≤ Mh . .
h
As h → 0, both mh and Mh tend to f(x).
Notes
Proving the Fundamental Theorem
Proof.
From §5.2 we have
∫ x+h
mh · h ≤ f(t) dt ≤ Mh · h
x
g(x + h) − g(x)
=⇒ mh ≤ ≤ Mh .
h
As h → 0, both mh and Mh tend to f(x).
.
.
Meet the Mathematician: James Notes
Gregory
Sco sh, 1638-1675
Astronomer and Geometer
Conceived transcendental
numbers and found evidence
that π was transcendental
Proved a geometric version of
1FTC as a lemma but didn’t take
it further
.
.
. 7
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8. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Meet the Mathematician: Isaac Notes
Barrow
English, 1630-1677
Professor of Greek, theology,
and mathema cs at Cambridge
Had a famous student
.
.
Meet the Mathematician: Isaac Notes
Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis Principia
Mathema ca published 1687
.
.
Meet the Mathematician: Gottfried Notes
Leibniz
German, 1646–1716
Eminent philosopher as well as
mathema cian
Contemporarily disgraced by the
calculus priority dispute
.
.
. 8
.

9. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Differentiation and Integration as Notes
reverse processes
Pu ng together 1FTC and 2FTC, we get a beau ful rela onship
between the two fundamental concepts in calculus.
Theorem (The Fundamental Theorem(s) of Calculus)
I. If f is a con nuous func on, then
∫
d x
f(t) dt = f(x)
dx a
So the deriva ve of the integral is the original func on.
.
.
Differentiation and Integration as Notes
reverse processes
Pu ng together 1FTC and 2FTC, we get a beau ful rela onship
between the two fundamental concepts in calculus.
Theorem (The Fundamental Theorem(s) of Calculus)
II. If f is a differen able func on, then
∫ b
f′ (x) dx = f(b) − f(a).
a
So the integral of the deriva ve of is (an evalua on of) the
original func on.
.
.
Notes
Outline
Recall: The Evalua on Theorem a/k/a 2nd FTC
The First Fundamental Theorem of Calculus
Area as a Func on
Statement and proof of 1FTC
Biographies
Differen a on of func ons defined by integrals
“Contrived” examples
Erf
Other applica ons
.
.
. 9
.

10. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Differentiation of area functions
Example
∫ 3x
Let h(x) = t3 dt. What is h′ (x)?
0
Solu on (Using 2FTC)
3x
t4 1
h(x) = = (3x)4 = 1
4 · 81x4 , so h′ (x) = 81x3 .
4 0 4
.
.
Notes
Differentiation of area functions
Example
∫ 3x
Let h(x) = t3 dt. What is h′ (x)?
0
Solu on (Using 1FTC)
∫ u
We can think of h as the composi on g ◦ k, where g(u) = t3 dt
0
and k(x) = 3x. Then h′ (x) = g′ (u) · k′ (x), or
h′ (x) = g′ (k(x)) · k′ (x) = (k(x))3 · 3 = (3x)3 · 3 = 81x3 .
.
.
Notes
Differentiation of area functions, in general
by 1FTC ∫
d k(x)
f(t) dt = f(k(x))k′ (x)
dx a
by reversing the order of integra on:
∫ ∫
d b d h(x)
f(t) dt = − f(t) dt = −f(h(x))h′ (x)
dx h(x) dx b
by combining the two above:
∫ (∫ ∫ 0 )
d k(x) d k(x)
f(t) dt = f(t) dt + f(t) dt
dx h(x) dx 0 h(x)
= f(k(x))k′ (x) − f(h(x))h′ (x)
.
.
. 10
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11. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Another Example
Example
∫ sin2 x
Let h(x) = (17t2 + 4t − 4) dt. What is h′ (x)?
0
Solu on
.
.
Notes
A Similar Example
Example
∫ sin2 x
Let h(x) = (17t2 + 4t − 4) dt. What is h′ (x)?
3
Solu on
.
.
Notes
Compare
Ques on
∫ 2 ∫ 2
d sin x d sin x
Why is (17t2 + 4t − 4) dt = (17t2 + 4t − 4) dt?
dx 0 dx 3
Or, why doesn’t the lower limit appear in the deriva ve?
Answer
.
.
. 11
.

12. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
The Full Nasty
Example
∫ ex
Find the deriva ve of F(x) = sin4 t dt.
x3
Solu on
No ce here it’s much easier than finding an an deriva ve for sin4 .
.
.
Notes
Why use 1FTC?
Ques on
Why would we use 1FTC to find the deriva ve of an integral? It
seems like confusion for its own sake.
Answer
Some func ons are difficult or impossible to integrate in
elementary terms.
Some func ons are naturally defined in terms of other
integrals.
.
.
Notes
Erf
∫ x
2
e−t dt
2
erf(x) = √
π 0
erf measures area the bell curve. erf(x)
We can’t find erf(x), explicitly,
but we do know its deriva ve: .
′ 2 x
erf (x) = √ e−x .
2
π
.
.
. 12
.

13. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Example of erf
Example
d
Find erf(x2 ).
dx
Solu on
.
.
Notes
Other functions defined by integrals
The future value of an asset:
∫ ∞
FV(t) = π(s)e−rs ds
t
where π(s) is the profitability at me s and r is the discount
rate.
The consumer surplus of a good:
∫ q∗
CS(q∗ ) = (f(q) − p∗ ) dq
0
where f(q) is the demand func on and p∗ and q∗ the
equilibrium price and quan ty.
.
.
Notes
Surplus by picture
consumer surplus
price (p)
producer surplus
supply
p∗ equilibrium
market revenue
demand f(q)
.
q∗ quan ty (q)
.
.
. 13
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14. . V63.0121.001: Calculus I
. Sec on 5.4: The Fundamental Theorem
. May 2, 2011
Notes
Summary
Func ons defined as integrals can be differen ated using the
first FTC: ∫
d x
f(t) dt = f(x)
dx a
The two FTCs link the two major processes in calculus:
differen a on and integra on
∫
F′ (x) dx = F(x) + C
.
.
Notes
.
.
Notes
.
.
. 14
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