7.1

Integral As Net Change

Quick Review

2

-

2

2

Find all values of (if any) at which the function changes

sign on the given interval.

1. cos 2 on 0,1

2. 5 6 on -5,5

3. on 0,

14. on -5,5

4

x

x

x

x x

e

x

x

4

3 ,2

positive always

2 ,1 ,1 ,2

What you’ll learn about Linear Motion Revisited General Strategy Consumption Over Time Net Change from Data Work

Essential QuestionHow can the integral be used to calculate netchange and total accumulation?

Example Linear Motion Revisited1. v(t) = 10 – 2t is the velocity in m/sec of a particle moving along the x-

axis when 0 < t < 9. Use analytical methods to:

a. Determine when the particle is moving to the right, to the left, and stopped.

b. Find the particle’s displacement for the given time interval.

c. If s(0) = 3, what is the particle’s final position?

d. Find the total distance traveled by the particle.

right moving is particle the,0When tv

50 t stopped is particle the,0When tv

5t left moving is particle the,0When tv

95 t

Example Linear Motion Revisited1. v(t) = 10 – 2t is the velocity in m/sec of a particle moving along the x-

axis when 0 < t < 9. Use analytical methods to:

b. Find the particle’s displacement for the given time interval.

9

0 210 dtt 9

0 210 tt

8190 00 9

It moves 9 units to the right in the first 9 seconds.

Example Linear Motion Revisited1. v(t) = 10 – 2t is the velocity in m/sec of a particle moving along the x-

axis when 0 < t < 9. Use analytical methods to:

c. If s(0) = 3, what is the particle’s final position?

dtt 210 Ctt 210 C 20010 3

3C

ts

310 2 ttts

399109 2 s 12

Which is the original position of 3 plus displacement of 9.

Example Linear Motion Revisited1. v(t) = 10 – 2t is the velocity in m/sec of a particle moving along the x-

axis when 0 < t < 9. Use analytical methods to:

d. Find the total distance traveled by the particle.

5

0 210 dtt

5

0 210 tt

2550 8190

25

9

5 210 tt

Total distance is 41 m.

9

5 210 dtt

2550

16 41

Strategy for Modeling with Integrals

1. Approximate what you want to find as a Riemann sum of values of a continuous function multiplied by interval lengths. If f (x) is the function and [a, b] the interval, and you partition the interval into subintervals of length x, the approximating sums will have the form

, with kk cxcf a point in the kth subinterval.

2. Write a definite integral, here to express the limit of these sums as the norm of the partitions go to zero.

,

b

adxxf

3. Evaluate the integral numerically or with an antiderivative.

Example Potato Consumption

2. From 1970 to 1980, the ratio of potato consumption in a particular country was C(t) = 2.2 + 1.1t million of bushels per year, with t being years since the beginning of 1970. How many bushels were consumed from the beginning of 1972 to the end of 1975? 6 ,2Step 1: Riemann sum

We partition [2, 6] into subintervals of length t and let tk be a time in the kth subinterval.The amount consumed during this subinterval is approximately C (tk ) t million bushels.

The consumption for [2, 6] is approximately C (tk ) t million bushels.

Example Potato Consumption

2. From 1970 to 1980, the ratio of potato consumption in a particular country was C(t) = 2.2 + 1.1t million of bushels per year, with t being years since the beginning of 1970. How many bushels were consumed from the beginning of 1972 to the end of 1975? 6 ,2Step 2: Definite Integral

The amount consumed from t = 2 to t = 6 is the limit of these sums as the norms of the partitions go to zero.

6

2 dttC

6

2 1.12.2 dtt

Step 3: Evaluate

Evaluate numerically, we obtain:

6 ,2 , ,1.12.2NINT tt 692.14 mil bushels

Work

.FdW

When a body moves a distance d along a straight line as a result of the action of a force of constant magnitude F in the direction of motion, the work done by the force is

The equation W = Fd is the constant – force formula for work.

The units of work are force x distance. In the metric system, the unit is the newton – meter or joule. In the U.S. customary system, the most common unit is the foot – pound.

Hooke’s Law for springs says that the force it takes to stretch or compress a spring x units from its natural length is a constant times x. In symbols: F = kx, where k, measured in force units per length, is a force constant.

Example A Bit of Work3. It takes a force of 6N to stretch a spring 2 m beyond its

natural length. How much work is done in stretching the spring 5 m from its natural length?

By Hooke’s Law, F(x) = kx. Therefore k = F(x) / x.

2/6k N/m 3 xxF 3Step 1: Riemann sum

We partition [0, 5] into subintervals of length x and let xk be any point in the kth subinterval.The work done across the interval is approximately F (xk ) x.The sum for [0, 5] is approximately F (xk ) x = 3xk x.

Step 2 and 3: Evaluate Definite Integral

5

0 3 dxx

5

0

2

2

3

x N/m

2

75

Pg. 386, 7.1 #1-29 odd