School

University of California, Berkeley **We aren't endorsed by this school

Course

MATH 16B

Subject

Mathematics

Date

Oct 9, 2023

Pages

4

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Definite Integral Explained And Rules
Page
1
of
4
To estimate the area under a positive function
(
)
f
x
from
x
a
=
to
x
b
=
(
A
), we can use a single rectangle
to do so. The height of the rectangle we use must be a
representative height of the function somewhere
within the interval
[
]
,
a b
.
Rectangle whose height is the largest value of
(
)
f
x
on the interval
[
]
,
a b
x
y
a
b
A
Rectangle whose height is the least value of
(
)
f
x
on the interval
[
]
,
a b
x
y
a
b
A
Rectangle using a height that is somewhere between the largest value of
(
)
f
x
and the least
value of
(
)
f
x
on the interval
[
]
,
a b
.
x
y
a
b
A
To improve the estimate of the area under the curve, we will need to use more rectangles to get a
more accurate approximation.
x
y
a
b
A
(
)
f
x

Definite Integral Explained And Rules
Page
2
of
4
Using more rectangles improves our estimate (figure #1). However, as we increase the number of
rectangles, we must also make sure that each rectangle becomes narrower, or else we will not
approach the true value of the area under the curve, as in figure #2.
x
y
a
b
A
x
y
a
b
A
Figure #1
Figure #2
In order to improve our estimate we must first
partition the interval
[
]
,
a b
into smaller subintervals.
That is we select a set of
x
-values
{
}
0,
1
2
,
,
,
n
x x
x
x
such that
[
]
[
]
[
]
0
1
1,
2
1
,
,
,
n
n
a b
x
x
x x
x
x
−
=
.
The partition of
[
]
,
a b
is
{
}
0,
1
2
,
,
,
n
x x
x
x
∆ =
.
Within each subinterval
[
]
1
,
i
i
x
x
+
we choose a
1
i
c
+
such that
1
1
i
i
i
x
c
x
+
+
≤
≤
The area of the
(
)
1
th
i
+
rectangle is
(
)(
)
1
1
i
i
i
f
c
x
x
+
+
−
.
The estimate for the area under the function from
x
a
=
to
x
b
=
is
given
by
(
)(
)
(
)
(
)(
)
(
)
(
)(
)
(
)
1
2
1
1
0
2
2
1
1
n
n
n
n
x
x
x
f
c
x
x
f
c
x
x
f
c
x
x
−
∆
∆
∆
−
+
−
+
+
−
(
)(
)
1
n
i
i
i
f
c
x
=
=
∑
.
In order to improve the estimate, we must create (1)
more rectangles, where (2) each rectangle becomes
narrower and narrower.
We define the "norm of delta", denoted
∆
to be the
width of the largest subinterval of the partition
∆
. To
create more and more rectangles of narrower and
narrower width, we let
0
∆ →
.
x
y
0
x
a
=
n
x
b
=
1
x
i
x
1
i
x
+
1
n
x
−
x
y

Definite Integral Explained And Rules
Page
3
of
4
If
(
)
f
x
is defined on the closed interval
[
]
,
a b
and
(
)(
)
0
1
lim
n
i
i
i
f
c
x
∆ →
=
∆
∑
exists, then we say that
(
)
f
x
is integrable on
[
]
,
a b
and
(
)(
)
(
)
0
1
lim
b
n
i
i
i
a
f
c
x
f
x dx
∆ →
=
∆
=
∑
∫
Where
(
)
b
a
f
x dx
∫
is read "The integral from
x
a
=
to
x
b
=
, of
(
)
f
x
, with respect to
x
."
a
is called the
lower bound
of the integral
b
is called the
upper bound
of the integral
(
)
f
x
is called the
integrand
dx
indicates that integration is done with respect to
x
.
Properties of Definite Integrals
I.
(
)
0
a
a
f
x dx
=
∫
II.
(
)
(
)
a
b
b
a
f
x dx
f
x dx
= −
∫
∫
"switch the direction, switch the sign"
III.
If
a
c
b
≤
≤
, then
(
)
(
)
(
)
b
c
b
a
a
c
f
x dx
f
x dx
f
x dx
=
+
∫
∫
∫
x
y
a
b
x
y
a
b
c
x
y
a
b
c
IV.
If
k
is a constant then
(
)
(
)
b
b
a
a
k
f
x dx
k
f
x dx
⋅
=
⋅
∫
∫
V.
(
)
(
)
(
)
(
)
b
b
b
a
a
a
f
x
g x dx
f
x dx
x dx
±
=
±
∫
∫
∫
VI.
If
f
is integrable and non-negative on
[
]
,
a b
, then
(
)
0
b
a
f
x dx
≤
∫
=
+

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