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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 particle starts at x  0 and moves along the x-axis with velocity v(t )  t
2
for time t  0. Where is the particle at t  3?
Graph v and partition the time interval into subintervals of length t. If you use
t  1/ 4, you will have 12 subintervals. The area of each rectangle approximates
the distance traveled over the subinterval. Adding all of the areas (distances)
gives an approximation to the total area under the curve (total distance traveled)
from t  0 to t  3.
Example Finding Distance
Traveled when Velocity Varies
Continuing in this manner, derive the area 1/ 4  m  for each subinterval and
2
i
add them:
1
9
25 49
81 121 169 225 289 361 441 529 2300












256 256 256 256 256 256 256 256 256 256 256 256 256
 8.98
Example Estimating Area Under the
Graph of a Nonnegative Function
Estimate the area under the graph of f ( x)  x sin 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  xi x 
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
2 1   1   1 
2 1   3   1 
2 1   5   1 
LRAM  0         1         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
1 2 23
2
2
1 1
2 1   3   1 
2 1   5   1 
2 1 
RRAM       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
2
2
2
2
2
 1   1   3   1   5   1   7   1   9   1   11   1 
MRAM                              
 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
is written as
n
a
i 1
i
a1 , a2 , a3 ,..., an
 a1  a2  a3  ...  an
is the index of summation
ai 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
 i
n
i 1
2
 
 
 



 1  12  1  2 2  1  32  1  ...  n 2  1
Section 5.1 – Estimating with Finite
Sums

Properties of Summation
n
1.
n
 ka  k  a
i
i 1
n
2.
i
i 1
n
n
 a  b    a   b
i 1
i
i
i 1
i
i 1
i
Section 5.1 – Estimating with Finite
Sums

Summation Formulas:
n
1.
2.
3.
4.
 c  cn
i 1
n
nn  1
i

2
i 1
nn  12n  1
i 

6
i 1
2
2
n


n
n

1
3
i


4
i 1
n
2
Section 5.1 – Estimating with Finite
Sums

Example:

10

10

2
3
i
i

1

i

 i
i 1

i 1
10
10
i 1
i 1
  i3   i
10 10  1 1010  1


4
2
100121 1011


4
2
 25121  511
2
 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 
n
n
 f m x  lim  f M x  lim S n 
i 1
i
n 
i 1
i
n 
ba
where x 
and f mi  and f M i  are the minimum
n
and maximum 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
Area  lim
n 
n
 f c x,
i 1
i
xi 1  ci  xi
(ci, f(ci))
ba
and x 
n
xi-1
xi