Unit 10.2

Copyright © 2011 Pearson, Inc.

10.2Limits and

Motion: The Area Problem

Copyright © 2011 Pearson, Inc. Slide 10.3 - 2

What you’ll learn about

Distance from a Constant Velocity Distance from a Changing Velocity Limits at Infinity The Connection to Areas The Definite Integral

… and whyLike the tangent line problem, the area problem has many applications in every area of science, as well as historical and economic applications.


Example Computing Area

Use the five rectangles in the figure to estimate the area of the region below the curve f(x) = x2 + 1 for x in the interval [0, 5].



The base of each approximating rectangle is 1 unit. The height is determined by the function evaluated at the left endpoint of the subintervals: 0, 1, 2, 3, and 4. The areas (base height) of each of the five approximating rectangles are:



The sum of the areas of the five rectangles is

1 + 2 + 5 + 10 + 17 = 35 square units

which is the desired estimate of the area.


Example Computing Distance Traveled

A car travels at an average rate of 56 miles per hour for 3 hours and 30 minutes. How far does the car travel?


Example Computing Distance Traveled

A car travels at an average rate of 56 miles per hour for 3 hours and 30 minutes. How far does the car travel?

The distance traveled is s, the time interval has

length t, and s

t is the average velocity. Therefore,

s s

tt 56 mph 3.5 hours 196 miles.


Limits at Infinity (Informal)

When we write "limx

f (x) L," we mean that f (x) gets

arbitrarily close to L as x gets arbitrarily large.


Let y f (x) be a continuous function over an interval

[a,b]. Divide [a,b] into n subintervals of length

x (b a) / n. Choose any value x1 in the first

subinterval, x2 in the second, and so on.

Compute f (x1), f (x

2), f (x

3),... f (x

n) multiply each value

by x and sum up the products. In sigma notation,

the sum of the products is f (xi) x

i1

n

f (xi)x

i1

a


Definite Integral

Let f be a function on [a,b] and let f (xi) x

i1

n

be

defined as above. The definite integral of f over [a,b],

denoted f (x)dxa

b

, is given by

f (x)dxa

b

limn

f (xi) x

i1

a

,

provided the limit exists.

If the limit exists, we say f is integrable on [a,b].


Quick Review

1. List the elements of the sequence ak k 2 for k 1,2,3,4.

Find the sum.

2. (k 1)k1

5

3. k 2

k1

5

4. A car travels at an average speed of 56 mph for 3 hours.

How far does it travel?

5. A pump working at 4 gal/min pumps for 3 hours.

How many gallons are pumped?


Quick Review Solutions

1. List the elements of the sequence ak k 2 for k 1,2,3,4.

1,4,9,16 Find the sum.

2. (k 1)k1

5

20 3. k 2

k1

5

55

4. A car travels at an average speed of 56 mph for 3 hours.

How far does it travel? 168 miles

5. A pump working at 4 gal/min pumps for 3 hours.

How many gallons are pumped? 720 gallons

Unit 10.2

Education

Transcript of Unit 10.2