MAT137 | Calculus! | Lecture 28
Today:
AntiderivativesFundamental Theorem of Calculus
Next:
More FTC(review v. 8.5 - 8.7)§5.7 Substitution(v. 9.1 - 9.4)
official website http://uoft.me/MAT137Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Properties of the Definite Integral
Let f and g be integrable functions, and let a,b, c be any real numbers.
1 [order of limits] ∫
b
af (x) dx = −∫
a
bf (x) dx
a b
+a b
−
2 [constant multiple] ∫
b
acf (x) dx = c ∫
b
af (x) dx
a ba b
y = cf (x)
y = f (x)
a ba b
y = cf (x)
y = f (x)
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
3 [sum] ∫
b
a(f (x) + g(x)) dx = ∫
b
af (x) dx + ∫
b
ag(x) dx
a b
y = f (x)
+
a b
y = g(x)
=
a ba b
y = f (x) + g(x)
4 [additivity] ∫
b
af (x) dx = ∫
c
af (x) dx + ∫
b
cf (x) dx
a bca b
y = f (x)
∫c
af (x)dx ∫
b
cf (x)dx
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Comparison Properties of the Integral
The following properties are true only if a ≤ b.
5 [integral of a non-negative function]
f (x) ≥ 0 on [a,b] Ô⇒ ∫
b
af (x) dx ≥ 0.
6 [domination]
If f (x) ≥ g(x) on [a,b] Ô⇒ ∫
b
af (x) dx ≥ ∫
b
ag(x) dx .
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Properties of the Definite Integral
Example 1
If
∫
1
−1f (x)dx = 2, ∫
5
1f (x)dx = 3, ∫
1
−1g(x)dx = 5, ∫
0
−1g(x)dx = 1,
find each of the following integrals, if possible:
(a) ∫1
−1(2f (x) + g(x)) dx
(b) ∫5
−1f (x) dx
(c) ∫1
0f (x) dx
(d) ∫1
0g(x) dx
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Properties of the Definite Integral
If f is continuous and f (x) < 0 for all x ∈ [a,b], then ∫ba f (x)dx
1 must be negative
2 might be 0
3 not enough information
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Properties of the Definite Integral
Let f be a continuous function on the interval [a,b].
True or False.
There exist two constants m and M, such thatm(b − a) ≤ ∫
ba f (x)dx ≤M(b − a)
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Antiderivatives
Definition (Antiderivative)
A function F is an antiderivative of a function f on an interval I if
F ′(x) = f (x) for all x in I .
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Antiderivatives
Theorem (General form of antiderivative)
If F is an antiderivative of f on an interval I , then the most generalantiderivative of f on I is
∫ f (x)dx = F (x) + C
where C ∈ R is an arbitrary constant.
Note: ∫ f (x)dx represents the collection of all functions whose
derivative is f (x).
Example 2
Find a function f such that f ′(x) = 3x2 and f (0) = 1.
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Antiderivatives
Example 3
Find the antiderivative of f (x) = (3x + 5)7.
Here’s the general strategy in the form of a flow diagram:
guess checkclose
not close
adjust checkcorrect
not quite correct
writemost
generalantiderivative
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Antiderivatives
Example 4
Find a function f (x) if f ′′(x) = sin x + ex − x2, and f (0) = 0, f ′(0) = 2.
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Indefinite Integral - Guess and Check
Example 5
Evaluate
∫ex
ex + 1dx
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Indefinite Integral - Guess and Check
Example 6
Evaluate
∫sin
√x
√x
dx
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Indefinite Integral - Guess and Check
Example 7
Evaluate
∫ sin(x) cos(x)dx
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
The fundamental theorem of calculus deals with functions of the form
g(x) = ∫x
af (t) dt,
where f is a continuous function on [a,b] and x varies between a and b.
For example, if f is non-negative, then g(x) can be interpreted as the areaunder the graph of f between a and x , where x varies from a to b. Youcan think of g as “the area so far” function.
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Area so Far Function
Let f (t) = t and a = 0, then the function
g(x) = ∫x
0tdt
represents the area under the curve in the picture. Thus,
g(x)
x
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Area so Far Function
Below is the graph of a function f .
-1 -0.5 0.5 1 1.5 2 2.5
1
2
3
4
Let g(x) = ∫x
0f (t)dt. Then for 0 < x < 2, g(x) is
1 increasing and concave up.
2 increasing and concave down.
3 decreasing and concave up.
4 decreasing and concave down.
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
Area so Far Function
Below is the graph of a function f .
-1 -0.5 0.5 1 1.5 2 2.5
1
2
3
4
Let g(x) = ∫x
0f (t)dt. Then
1 g(0) = 0, g ′(0) = 0 and g ′(2) = 0
2 g(0) = 0, g ′(0) = 4 and g ′(2) = 0
3 g(0) = 1, g ′(0) = 0 and g ′(2) = 1
4 g(0) = 0, g ′(0) = 0 and g ′(2) = 1
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
FTC | “Area so Far” function
The fundamental theorem of calculus deals with functions of the form
g(x) = ∫x
af (t) dt,
where f is a continuous function on [a,b] and x varies between a and b.Remark: t is a ”dummy variable”
For example, if f is non-negative, then g(x) can be interpreted as the areaunder the graph of f between a and x , where x varies from a to b.You can think of g as “the area so far” function.
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
FTC | “Area so Far” function
Let f (t) = t and a = 0, then the function
g(x) = ∫x
0t dt
represents the area under the curve in the picture.
g(x)
x
g(x) = ∫x
0t dt = 1
2x ⋅ x =1
2x2.
What is g ′(x)?
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
FTC | “Area so Far” function
Let f (t) = t and a = 0, then the function
g(x) = ∫x
0t dt
represents the area under the curve in the picture.
g(x)
x
g(x) = ∫x
0t dt = 1
2x ⋅ x =1
2x2.
What is g ′(x)?
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
FTC | “Area so Far” function
Let f (t) = t and a = 0, then the function
g(x) = ∫x
0t dt
represents the area under the curve in the picture.
g(x)
x
g(x) = ∫x
0t dt = 1
2x ⋅ x =1
2x2.
What is g ′(x)?Beatriz Navarro-Lameda L0601 MAT137 16 January 2018
FTC | Part I
Theorem (Fundamental Theorem of Calculus, Part 1 [FTC1])
If f is continuous on an interval [a,b], then the function g defined by
g(x) = ∫x
af (t) dt for a ≤ x ≤ b
is continuous on [a,b] and differentiable on (a,b).Moreover, g ′(x) = f (x).
Roughly speaking, this says the following: when f is continuous, if we firstintegrate and then differentiate, we get f back.
Beatriz Navarro-Lameda L0601 MAT137 16 January 2018