1. Drive activity
• Please submit your drive activities.

2. Sec. 4.1: Antiderivatives

3. Uses of integration
1.
2.
3.
4.
5.
“Undoes” differentiation.
Finds the area under a curve.
Finds the volume of a solid.
Finds the center of mass.
Finds s(t) given a(t) or v(t).
*** and many more….

4. The Uniqueness of Antiderivatives
2
Suppose f x
3x , find an antiderivative of f. That is,
2
find a function F(x) such that F ' x
3x .
F x
x
3
Using the Power Rule in
Reverse
Is this the only function whose derivative is 3x2?
H x
H' x
x
3
5
3x 2
K x
x
3
11
3x 2
K' x
M x
M' x
x
3
3x 2
There are infinite functions whose derivative is 3x2
whose general form is:
C is a constant real
G x
3
x
C
number (parameter)

5. When we find antiderivatives and add the constant C,
we are creating a family of curves for each value of C.
x3 C
C=0
C=1
C=3

6. Indefinite Integral
f ( x)dx
Integral
Sign
Integrand
F ( x) C
The
Indefinite
Integral
Variable of
Integration
The constant
of Integration

7. Indefinite Integral
The indefinite integral gives a family of functions!
(Not a value)
The indefinite integral always has a constant!
Vs.
The definite integral (later) gives a numerical
value.

8. Summary
Integration is the “inverse” of differentiation.
d
F ( x) dx
dx
F ( x) C
Differentiation is the “inverse “ of integration.
d
dx
f ( x)dx
f ( x)

9. Basic Integration Rules
0dx
C
kdx
kx C
k f ( x )dx
k f ( x )dx

10. Basic Integration Rules
f ( x ) g ( x ) dx
f ( x )dx
n 1
n
x dx
x
C
n 1
Power Rule
g ( x )dx

11. Basic Integration Rules
cos xdx
sin xdx
2
sec xdx
sin x C
cos x C
tan x C

12. Basic Integration Rules
2
csc xdx
sec x tan xdx
csc x cot xdx
cot x C
sec x C
csc x C

13. Examples
Find each of the following indefinite integrals.
5
a.
x dx
b.
5 x3 dx
x
sin x dx
c.
1
6
d.
1
x
6
C
cos x C
5
4
x4
C
dx 2 x C

14. Application problem
• A ball is thrown upward with an initial
velocity of 64 ft/ sec from an initial height
of 80 feet.
a. Find the position function, s(t)
b. When does the ball hit the ground?

15. Example
A particle moves along a coordinate axis in such a way that
3
its acceleration is modeled by a t
2t for time t > 0.
If the particle is at s = 5 when t = 1 and has velocity v = - 2
at this time, where is it when t = 4?
Integrate the acceleration to find velocity:
v t
2t
3
dt
2 t
3
1
3 1
2
dt
t
3 1
t
2
t
C
2
C
Use the Initial Condition to find C for velocity:
2
1
2
C
C
v t
1
1
Integrate the Velocity to find position:
s t
t
5
1
2
1
Answer the
Question:
1 dt
t 2 dt
1
2 1
1 dt
t
2 1
1t C t
Use the Initial Condition to find C for position:
1 C
s 4
C
4
1
4 5
5
1.25
s t
t
1
1
t C
t 5