Section 5.1 1

Section 5.1: Areas and Distances

Section 5.1 2

Section 5.1: Area and Distances

• Objective– Understand the basic properties of area– Understand the area of basic shapes– Understand rudimentary area approximations– Be able to calculate the area under a curve

Section 5.1 3

Properties of Area

Section 5.1 4

Properties of Area: Typed

1. The area of a plane region is a nonnegative real number

2. Area is measured in square units

3. Congruent regions have equal areas

4. The area of the union of two regions that overlap only in a line segment equals the sum of the areas of the two regions

5. If one region is contained in a second, then the area of the first is less than or equal to that of the second

Section 5.1 5

Basic Area Formulas

A=A1+A2+A3+A4+…+An

Polygon

A=Triangle

A=bhParallellogram

A=lwRectangle

l

w

12

bh

h

b

b

h

A1A2

A3

A4A5

Section 5.1 6

Sigma Notation

Section 5.1 7

Sigma Notation: Typed

The sum of n terms is written as:1 2, 3,, ..., na a a a

1 2 31

...,n

i ni

a a a a a=

= + + +∑

where i is the index of summation, is the i’th term of the sum ia

1 = lower bound of the sum, and n is the upper bound

Section 5.1 8

Sigma Notation: Examples

8 times

Section 5.1 9

Sigma Notation: Example: Typed6

1

62

2

1

8

1

1 2 3

.

.

. ( )

. 7

. , , ,...,

ii

j

n

ii

i

n

a a

b j

c f x x

d

e Average of n numbers a a a a

=

=

=

=

∆

∑

∑

∑

∑

Section 5.1 10

Summation Formulas

1 1

n n

i ii i

ka k a= =

=∑ ∑

( )1 1 1

n n n

i i i ii i i

a b a b= = =

+ = +∑ ∑ ∑

( )1 1 1

n n n

i i i ii i i

a b a b= = =

− = −∑ ∑ ∑

Section 5.1 11

Area

We already know how to find the area of rectangles, triangles, polygons, circles, etc. How do we find the area of a region S that lies under the curve y = f(x)

from a to b?

Strategy:

1) Divide the interval [a,b] into n intervals of equal width

2) Construct n rectangles from these intervals in such a way that the width of each rectangle will be the width of the interval and such that the height of each rectangle will be f evaluated the right or left endpoint of the interval used to construct the rectangle.

3) Sum the areas of the n rectangles

4) This gives an approximation to the area of the rectangle

5) Take Limits

Section 5.1 12

Example 1: y = 3x+1Estimate the area under f(x) = 3x + 1 over the interval [1 , 3].Divide the interval [1 , 3] into 4 equal subintervals and calculate the area of the corresponding circumscribed polygon.

Solution:

1 2 3

1

2

3

4

5

6

7

8

9

10

Section 5.1 13

Example 1: Divide The Interval [1,3]Divide the interval [1 , 3] into 4 equal subintervals

Solution:

1 2 3

1

2

3

4

5

6

7

8

9

10

f(x) =3x + 1

Section 5.1 14

4 Subintervals

1 33/2

2 5/2

3 3 5 51, , , 2 , 2, , ,32 2 2 2

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

intLet width of each sub erval∆ =

3 1 14 2

Then −∆ = =

Section 5.1 15

Calculate the AreaCalculate the area of the corresponding circumscribed polygon. Called this approximation an Upper Sum.

1 2 3

123456789

103 1 1

4 2x −∆ = =

The sum of the reas of the 4 rectangles is:A

1 11 14 17 20 312 2 2 2 2 2

A ⎛ ⎞= + + + =⎜ ⎟⎝ ⎠

4Let R = approximated area using right endpoints

( )1 3 1 1 5 12 (3)2 2 2 2 2 2

A f f f f⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( ) 3 1f x x= +

Section 5.1 16

ExampleFind an Upper Sum the area under y= x+1, between 0 and 2, when n=8

Solution:

1 2

1

2

3

y = x +

11Since 8, 4

n x= ∆ =

1The area of each rectangle is ( )4

f x

41 1 2 3 4 5 6 7 84 4 4 4 4 4 4 4 4

R f f f f f f f f⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦1 5 6 7 8 9 10 11 12 174 4 4 4 4 4 4 4 4 4⎛ ⎞= + + + + + + + =⎜ ⎟⎝ ⎠

0

Section 5.1 17

Summary: Definition of Area

The area A of the region S that lies under the graph of a continuous function f is the limit of the sum of the areas of approximatingrectangles. In symbols:

[ ]1lim lim ( ) ( 2) ... ( )n nn nA R f x x f x x f x x→∞ →∞= = ∆ + ∆ + + ∆

1

lim ( )n

in i

A f x x→∞

=

= ∆∑

OR[ ]0 1lim lim ( ) ( 2) ... ( )n nn nA L f x x f x x f x x−→∞ →∞= = ∆ + ∆ + + ∆

11

lim ( )n

in i

A f x x−→∞=

= ∆∑

intnR right endpo s=

intnL right endpo s=

Section 5.1 18

Example 2:Find the area of the region under the curve over the interval [0,1].21 1

2y x= +

-1 1 2

-1

1

2

21 12

y x= +

To Be Continued