AP CALCULUS AB Chapter 5: The Definite Integral Section 5.1: Estimating with Finite Sums
Dec 13, 2015
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 whyLearning 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?
2 5
75mph
t
v(t)
TDT = Area under line = 3(75) = 225 miles
Section 5.1 – Estimating with Finite Sums Distance Traveled at Non-Constant Velocity:
75
2 5 8t
v(t)
Total Distance Traveled = Area of geometric figure = (1/2)h(b1+b2) = (1/2)75(3+8) = 412.5 miles
Example Finding Distance Traveled when Velocity Varies
2A particle starts at 0 and moves along the -axis with velocity ( )
for time 0. Where is the particle at 3?
x x v t t
t t
Graph and partition the time interval into subintervals of length . If you use
1/ 4, you will have 12 subintervals. The area of each rectangle approximates
the distance traveled over the subint
v t
t
erval. Adding all of the areas (distances)
gives an approximation to the total area under the curve (total distance traveled)
from 0 to 3.t t
Example Finding Distance Traveled when Velocity Varies
2
Continuing in this manner, derive the area 1/ 4 for each subinterval and
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 2568.98
im
Example Estimating Area Under the Graph of a Nonnegative Function
2Estimate the area under the graph of ( ) sin from 0 to 3.f x x x x x
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 secLower Sum = Area ofinscribed = s(n)
Upper Sum = Areaof circumscribed= S(n)
Midpoint Sum
n
ii xxfA
1
sigma = sum y-value at xi
width of region
nSns region of Area
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)
1 2 3
y=x2
LRAM = Left-hand Rectangular Approximation Method
= sum of (height)(width) of each rectangleheight is measured on left side of
each rectangle
875.6
2
1
2
5
2
12
2
1
2
3
2
11
2
1
2
1
2
10
22
22
22
LRAM
Section 5.1 – Estimating with Finite Sums Rectangular Approximation Method (cont.)
y=x2
RRAM = Right-hand RectangularApproximation Method
= sum of (height)(width) of eachrectangle
height is measured on right sideof rectangle
375.11
2
13
2
1
2
5
2
12
2
1
2
3
2
11
2
1
2
1 22
22
22
RRAM
1 2 3
Section 5.1 – Estimating with Finite Sums Rectangular Approximation Method (cont.)
y=x2
1 2 3
MRAM = Midpoint RectangularApproximation Method
= sum of areas of each rectangleheight is determined by the heightat the midpoint of each horizontal region
9375.8
2
1
4
11
2
1
4
9
2
1
4
7
2
1
4
5
2
1
4
3
2
1
4
1222222
MRAM
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
is the index of summationis the ith term of the sum
and the upper and lower bounds of summation are n and 1 respectively.
naaaa ,...,,, 321
n
ini aaaaa
1321 ...
iai
Section 5.1 – Estimating with Finite Sums Examples:
1...1312111
54321
2222
1
2
5
1
ni
i
n
i
i
Section 5.1 – Estimating with Finite Sums Properties of Summation
1.
2.
n
i
n
iii akka
1 1
n
i
n
i
n
iiiii baba
1 1 1
Section 5.1 – Estimating with Finite Sums Summation Formulas:
1.
2.
3.
4.
n
i
cnc1
n
i
nni
1 2
1
n
i
nnni
1
2
6
121
n
i
nni
1
223
4
1
Section 5.1 – Estimating with Finite Sums Example:
3080
115121252
1110
4
1211002
11010
4
11010
1
22
10
1
10
1
3
10
1
10
1
32
i i
i i
ii
iiii
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 of both the lower and upper sums exist and are equal to each other
l.subinterva on the of valuesmaximum and
minimum theare and and where
limlimlimlim
th
1 1
if
Mfmfn
abx
nSxMfxmfns
ii
n
i
n
in
in
inn
n
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
abx
xcxxcf iii
n
ii
n
and
,lim Area 11 (ci, f(ci))
xi-1 xi