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Section 4.3
In the 1800’s, the German mathematician, Georg
Riemann, used the limit of a sum to define the
area of a region in a plane.
On the AP Exam the questions will ask for a left
Riemann sum (left endpoint sum) or a right
Riemann sum (right endpoint sum) or a midpoint
Riemann sum (midpoint sum).
Riemann Sum
n
 f  x    x 
i 1
i
i
n = # of rectangles (partitions)
f  x
f (xi) = height of each rectangle
∆xi = width of each rectangle
ba
n
a x
b
Consider the following limit:
n
lim  f  xi    xi  = L → Area under the curve
x0
i 1
f  x
a x
b
Definition of a Definite Integral
If f is defined on the closed interval [a, b] and
n
lim  f  xi    xi 
x0
i 1
exists, then f is integrable (can be integrated) on
[a, b] and the limit is denoted by
n
 f  x  dx  lim  f  x    x 
b
a
x0
i 1
i
i
This symbol means the sum from a to b.
Vocabulary
The limit is called the definite integral. This is
always a number.
The number “a” is called the lower limit of
integration.
The number “b” is called the upper limit of
integration.
The function “f (x)” is called the integrand.
The symbol “dx” is Δx.
n
 f  x  dx  lim  f  x    x 
b
a
x0
i 1
i
i
Area under a curve can be represented using a
definite integral.
f (x)
b
 f  x  dx
a
a
b
Examples
1.

4
0
2 dx  8
Area of rectangle
=L∙W
=2∙4
2
1
1
2
3
4
2.

2
0
2x dx  4
Area of rt. ∆
1
 bh
2
1
  2  4 
2
4
3
2
1
1
2
3
4
3.

2
2
4  x dx  2
2
3
Area of semicircle
1 2
 r
2
1
2
    2 
2
2
1
-2
-1
1
2
4.

1
2
0
1
2x dx  2 x dx   2 x dx  -3
2
Area I is a negative #.
Area II is a positive #.
1

AI     4  2   4
2

1
AII   2 1  1
2
2
0
1
II
-2
-1
I
1
-1
-2
-3
-4
Properties of Definite Integrals
Definitions of Two Special Definite
Integrals
1.
 f  x  dx  0
a
a
16

4
4
x dx  0
2
12
8
4
-5
-4
-3
-2
-1
1
2
3
4
5
No area under the curve
 f  x  dx    f  x  dx
a
2.
b
b

0
4
a
where a < b
4
3x dx    3x dx
12
0
1
   4 12 
2
 24
8
4
1
2
3
4
5
Additive Property of Definite
Integrals
 f  x  dx   f  x  dx   f  x  dx
3.
 x
4
0
3
c
b
c
a
a
b
 1 dx    x  1 dx    x  1 dx
2
0
3
4
2
3
4.

b
a
kf  x  dx  k  f  x  dx
b
a


0
0
  4sin x  dx  4
5.
sin x dx
 f  x   g  x   dx   f  x  dx   g  x  dx
a
a
a

b
 x
4
1
b
2
 2 x  dx 

4
1
b
4
x dx  2  x dx
2
1
Examples
1.

1
1
0
1
1
0
x dx   x dx   x dx 1
1
1
 11  11
2
2
1 1
 
2 2
1
-1
1

Given:
1
Find:

 x
2
3
1
3
3
1
3
26 3
x dx  ;  x dx  4;  dx  2
1
3 1
2
2

x
  4 x  3 dx
 4 x  3 dx    x dx  4  x dx  3 dx
3
1
2
3
1
3
1
26
   4  4  3 2 
3
26
   16  6
3
26
26 30 4
   10   

3
3
3 3
AP Free Response Question
Let f be a function that is twice differentiable for all
real numbers. The table below gives values of f for
selected points in the closed interval 2 ≤ x ≤ 13.
Use a left Riemann sum with subintervals indicated
by the data in the table to approximate

13
2
f  x .
Show the work that leads to your answer.

13
2
f  x   f  2  x1  f  3 x2  f  5  x3  f  8  x4
 1 3  2   4  5  3   2 8  5   3 13  8 
 11  4  2    2  3  3  5 
 1  8  6  15
 18