AP CALCULUS AB: Integration
section 5.1
The Definite Integral
∫
b
a
f (x)dx
“The definite integral from a to b for f(x)”
The signed area (+ or -) bounded by
The x-axis, f(x), x=a, and x=b.
The Definite Integral
∫
b
a
f (x)dx
The signed area (+ or -) bounded by
The x-axis, f(x), x=a, and x=b.
The Definite Integral
e
∫
d
∫ f (x)dx =
c
d
∫
c
+
f (x)dx = -
d
e
f (x)dx + ∫ f (x)dx = ?
d
AN IMPROPER INTEGRAL
An improper integral for:
∫
b
a
f (x)dx
is when a>b
for example:
1
∫x
5
3
dx
What do you do with
1
∫ x dx
to make it proper???
3
5
You rewrite it this way:
1
∫x
5
5
3
dx = − ∫ x dx
3
1
Property of a definite integral:
b
a
a
b
∫ f (x)dx = − ∫ f (x)dx
INTERPRETING THE DEFINITE
INTEGRAL
Evaluate:
10
∫
0
f (x)dx = 600
miles
6
180
g(x)dx
=
gallons
∫
0
Water is running into a pool and at
some point water is being drained
out of the pool.
Describe what happened from t=0 to t=5.
Properties of the definite integral
b
>
g(x)dx
∫
a
b
∫
a
f (x)dx
Properties of the definite integral
c
∫
a
b
b
c
a
f (x)dx + ∫ f (x)dx = ∫ f (x)dx
Evaluate the following:
4
1.
g(x)dx
=
8
∫
0
0
2.
g(x)dx
=
-8
∫
−4
4
3.
∫ g(x)dx = 6
−2
4
4.
∫ g(x)dx =
−4
0
Evaluate the following:
1
5.
g(x)dx
=
0
∫
1
Property of a definite integral
a
∫ f (x)dx = 0
a
6
6.
∫ g(x)dx = 18
0
0
7.
∫ g(x)dx = -18
6
Evaluate the following:
0
∫ g(x)dx = -8
8.
−4
0
8
|
g(x)
|
dx
=
∫
9.
10.
6
6
0
0
−4
218=36
2g(x)dx
=
2
g(x)dx
=
∫
∫
Property of a definite integral:
**If c is a constant
a
a
b
b
∫ cf (x)dx = c ∫ f (x)dx