AP CALCULUS AB
Chapter 5:
The Definite Integral
Section 5.1:
Estimating with Finite Sums
What you’ll learn about
Distance Traveled
 Rectangular Approximation Method (RAM)
 Volume of a Sphere
 Cardiac Output

… and why
Learning about estimating with finite sums
sets the foundation for understanding
integral calculus.
Section 5.1 – Estimating with Finite
Sums

Distance Traveled at a Constant Velocity:
A train moves along a track at a steady rate of
75 mph from 2 pm to 5 pm. What is the total
distance traveled by the train?
v(t)
75mph
TDT = Area under line
= 3(75)
= 225 miles
2
5
t
Section 5.1 – Estimating with Finite
Sums

Distance Traveled at Non-Constant Velocity:
v(t)
75
Total Distance Traveled =
Area of geometric figure
= (1/2)h(b1+b2)
= (1/2)75(3+8)
= 412.5 miles
2
5
8
t
Example Finding Distance
Traveled when Velocity Varies
A p article starts at x  0 an d m o ves alo n g th e x -ax is w ith velo city v ( t )  t
2
fo r tim e t  0 . W h ere is th e p article at t  3?
G raph v and partition the tim e interval into subintervals of length  t . If you us e
 t  1 / 4, you w ill have 12 subintervals. T he area of each rectangle approxim ates
the distance traveled over the subint erval. A dding all of the areas (distanc es)
gives an approxim ation to the total area under the curve (total distance travele d)
from t  0 to t  3.
Example Finding Distance
Traveled when Velocity Varies
C ontinuing in this m anner, derive the ar ea  1 / 4   m
 for each subinterval and
2
i
add them :
1

256
 8.98
9
256

25
256

49
256

81
256

121
256

169
256

225
256

289
256

361
256

441
256

529
256

2300
256
Example Estimating Area Under the
Graph of a Nonnegative Function
E stim ate the area under the graph of f ( x )  x s in x from x  0 to x  3.
2
Applying LRAM on a graphing calculator using 1000 subintervals,
we find the left endpoint approximate area of 5.77476.
Section 5.1 – Estimating with Finite
Sums

Rectangular Approximation Method
15
5 sec
Lower Sum = Area of Upper Sum = Area
inscribed = s(n)
of circumscribed= S(n)
s  n   Area of region  S  n 
n
A
 f  x  x 
i
i 1
sigma = sum
Midpoint Sum
width of region
y-value at xi
LRAM, MRAM, and RRAM approximations to the area
under the graph of y=x2 from x=0 to x=3
Section 5.1 – Estimating with Finite
Sums

Rectangular Approximation Method (RAM)
(from Finney book)
y=x2
LRAM = Left-hand Rectangular
Approximation Method
= sum of (height)(width) of each
rectangle
height is measured on left side of
each rectangle
1 2 3
2
2
2
1 1 1
3 1
5 1
2 1 
2 1 
LRAM  0          1           2        
2 2 2
2 2 2
2 2 2
2
 6 . 875
Section 5.1 – Estimating with Finite
Sums

Rectangular Approximation Method (cont.)
y=x2
RRAM = Right-hand Rectangular
Approximation Method
= sum of (height)(width) of each
rectangle
height is measured on right side
of rectangle
RRAM
1 2 23
2
2
1 1
3 1
5 1
2 1 
2 1 
2 1 
      1           2          3   
2 2
2 2 2
2 2 2
2
 11 . 375
Section 5.1 – Estimating with Finite
Sums

Rectangular Approximation Method (cont.)
y=x2
MRAM = Midpoint Rectangular
Approximation Method
= sum of areas of each rectangle
height is determined by the height
at the midpoint of each horizontal region
1 2 3
2
MRAM
2
2
2
2
2
 1   1   3   1   5   1   7   1   9   1   11   1 
                  
4 2 4 2 4 2 4 2 4 2  4  2
 8 . 9375
Section 5.1 – Estimating with Finite
Sums

Estimating the Volume of a Sphere
The volume of a sphere can be estimated
by a similar method using the sum of the
volume of a finite number of circular
cylinders.
definite_integrals.pdf (Slides 64, 65)
Section 5.1 – Estimating with Finite
Sums

Cardiac Output problems involve the
injection of dye into a vein, and
monitoring the concentration of dye over
time to measure a patient’s “cardiac
output,” the number of liters of blood the
heart pumps over a period of time.
Section 5.1 – Estimating with Finite
Sums

See the graph below. Because the
function is not known, this is an
application of finite sums. When the
function is known, we have a more
accurate method for determining the area
under the curve, or volume of a
symmetric solid.
Section 5.1 – Estimating with Finite
Sums

Sigma Notation (from Larson book)
The sum of n terms a1 , a 2 , a 3 ,..., a n
is written as
n
a
i
 a 1  a 2  a 3  ...  a n
i 1
is the index of summation
a i is the ith term of the sum
and the upper and lower bounds of summation
are n and 1 respectively.
i
Section 5.1 – Estimating with Finite
Sums

Examples:
5
 i  1 2  3 4  5
i 1
n
 i
i 1
2
 
 
 


 1  1  1  2  1  3  1  ...  n  1
2
2
2
2

Section 5.1 – Estimating with Finite
Sums

Properties of Summation
n
1.
 ka
i 1
n
2.
 a
i 1
i
n
i
 k  ai
i 1
 bi  
n
n
a b
i
i 1
i 1
i
Section 5.1 – Estimating with Finite
Sums

Summation Formulas:
n
1.
2.
 c  cn
i 1
n
i
i 1
3.
4.
n
i
2

i 1
n
i
i 1
3

n n  1
2
n  n  1  2 n  1 
6
2
2
n n  1
4
Section 5.1 – Estimating with Finite
Sums

Example:
10
10
 
  i
i i 1 
2
i 1

i 
3
i
i 1
10 10  1 
2
4


10
i 1

i
i 1
10

3
100 121 
4

2

10 10  1 
10 11 
2
 25 121   5 11 
 3080
2
Section 5.1 – Estimating with Finite
Sums

Limit of the Lower and Upper Sum
If f is continuous and non-negative on the
interval [a, b], the limits as n  
of both
the lower and upper sums exist and are equal to
each other
lim s  n   lim
n 
n 
where  x 
n
 f  m  x  lim  f  M  x  lim S  n 
i
i 1
ba
n
and maximum
n
and
n 
i
i 1
n 
f  m i  and f  M i  are the minimum
values of f on the i
th
subinterva
l.
Section 5.1 – Estimating with Finite
Sums

Definition of the Area of a Region in the Plane
Let f be continuous an non-negative on the
interval [a, b]. The area of the region bounded
by the graph of f, the x-axis, and the vertical
lines x=a and x=b is
n
Area  lim
n 
and  x 
 f c  x ,
i
i 1
x i 1  c i  x i
(ci, f(ci))
ba
n
xi-1
xi