AP CALCULUS AB

Chapter 5:The Definite Integral

Section 5.1:Estimating with Finite Sums

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

Greenfield Village, Michigan

EX: pg274 #13,14,15,22In Class: pg 274 #16, 20, 23 HW: Textbook pg 274 #16, 17-25 odds 25 xtra credit

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Life is just a minute—only sixty seconds in it.Forced upon you—can't refuse it.Didn't seek it—didn't choose it.But it's up to you to use it.You must suffer if you lose it.Give an account if you abuse it.Just a tiny, little minute,But eternity is in it!

By Dr. Benjamin Elijah Mays, Past President of Morehouse College

Calculus Date: 2/7/2014 ID Check Objective: SWBAT complete assessment on Optimization and related rates problems.Do Now: Quiz Section 5.4 and 5.6Turn in SM and Textbook Homework on these sectionsHW Requests:

HW: Textbook QR 6.1 #1-10Be careful of units and Conversions Read Section 6.1Bring your calculators.Casio- please see if you have RAM programAnnouncements: Saturday SessionsRm 241 10-12:50

Life Is Just A Minute

Life is just a minute—only sixty seconds in it.Forced upon you—can't refuse it.Didn't seek it—didn't choose it.But it's up to you to use it.You must suffer if you lose it.Give an account if you abuse it.Just a tiny, little minute,But eternity is in it!

By Dr. Benjamin Elijah Mays, Past President of Morehouse College

Section 6.1 – 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 6.1 – 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.

Circumscribed rectangles are all above the curve:

Inscribed rectangles are all below the curve:

Section 6.1 – 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

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.

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 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