6036: Area of a Plane Region AB Calculus. Accumulation vs. Area Area is defined as positive. The...
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Transcript of 6036: Area of a Plane Region AB Calculus. Accumulation vs. Area Area is defined as positive. The...
6036: Area of a Plane Region
AB Calculus
Accumulation vs. Area
Area is defined as positive.
The base and the height must be positive.
Accumulation can be positive, negative, and zero.
h = always Top minus Bottom (Right minus Left)
๐ โ0=h
h=0 โ ๐
AreaDEFN: If f is continuous and non-negative on [ a, b ], the
region R, bounded by f and the x-axis on [ a,b ] is
Remember the 7 step method.
b = Perpendicular to the axis!
h = Height is always Top minus Bottom!
( )b
aTA f x dx
a b
( ) 0
lim ( )
b x
h f x
TA f x dx
Area of rectangle
[๐ ,๐ ]
Ex:
Find the Area of the region bounded by the curve,
and the x-axis bounded by [ 0, ]
siny x x
๐=โ ๐ฅ [ 0 ,๐ ]h=(๐ฅ+sin ๐ฅ ) โ0๐ด= (๐ฅ+sin ๐ฅ ) โ๐ฅ
lim๐โ โ
โ (๐ฅ+sin ๐ฅ ) โ ๐ฅ
๐ด=0
๐
(๐ฅ+sin ๐ฅ )๐๐ฅ
๐ด= ๐ฅ2
2โcos ๐ฅ|๐0
๐ด= ๐ 2
2โ (โ1 ) โ ( 0โ1 )
๐ด= ๐ 2
2+2
Ex:
Find the Area of the region bounded by the curve,
and the x-axis bounded by [ -1, 1 ]
3 2y x
๐=โ ๐ฅ [โ 1,1 ]h=0 โ(โ 3โ๐ฅโ2)
lim๐โ โ
โ ( 3โ๐ฅ+2 )
โ1
1
( 3โ๐ฅ+2 )๐๐ฅ
โ1
1 ( (๐ฅ )13 +2)๐๐ฅ
34
(๐ฅ )43 +2 ๐ฅ| 1
โ 1
( 34โ1+2)โ( 3
4โ (1 ) โ2)=( 3
4+2)โ( 3
4โ2)
34
โ34+2+2=4
Area between curves
REPEAT: Height is always Top minus Bottom!
( ) ( )b
aTA f x g x dx
a b
f (x)
g (x)1 ( )b
R aA f x dx
2 ( )b
R aA g x dx
Height of rectangle
Area between curves
The location of the functions does not affect the formula.
( ) ( )b
aTA f x g x dx
a b
Both aboveh=f-g
One above one belowh=(f-0)+(0-g)h=f-g
Both belowh=(0-g)-(0-f)h=f-g
<Always Top-bottom>
Area : Method:
Find the area bounded by the curves and
on the interval x = -1 to x = 2
2 1y x
2y x
๐=โ ๐ฅ [โ 1,2 ]h=(๐ฅ2+1 ) โ (๐ฅโ 2 )
h=๐ฅ2โ๐ฅ+3
lim๐โ โ
โ (๐ฅ2โ๐ฅ+3 ) โ ๐ฅ
โ1
2
(๐ฅ2โ๐ฅ+3 )๐๐ฅ
๐ฅ3
3โ๐ฅ2
2+3 ๐ฅ| 2
โ1
( 83
โ42+3 (2 ))โ(โ1
3โ
12+3 (โ 1 ))
93
โ32+9=3+9 โ1.5=10.5
Area : Example (x-axis):
Find the area bounded by the curves and2( ) 4f x x 2( ) 2g x x
๐=โ ๐ฅ [โโ3 ,โ3 ]
4 โ๐ฅ2=๐ฅ2โ 2
6=2 ๐ฅ2
3=๐ฅ2
ยฑโ3=๐ฅ
h=( 4 โ๐ฅ2 ) โ (๐ฅ2 โ2 )h=6 โ 2๐ฅ2
lim๐โ โ
โ (6โ 2๐ฅ2 ) โ ๐ฅ
โ โ3
โ3
(6 โ 2๐ฅ2 )๐๐ฅ
6 ๐ฅโ 2( ๐ฅ3
3 )| โ3โโ3
6 (โ3 ) โ 23
(โ3 )3 โ(โ 6โ3โ( 23 ) (โโ3
3 ))6 โ3 โ 2โ3+6โ3 โ 2โ3=8โ3
Area: Working with y-axis
Area between two curves.
The location of the functions does not affect the formula.
When working with y-axis, height is always Right minus Left.
( ( ) ( ))
lim ( ( ) ( ))
b y
h h y k y
TA h y k y y
h (y)
k (y)
a
b
( ( ) ( ))b
aTA h y k y dy
Perpendicular to y-axis!
Area : Example (y-axis):Find the area bounded by the curves
and
2 2y x2 2y x
๐ฅ= ๐ฆ2
2
๐ฅ=๐ฆ+2
2
Perpendicular to y-axis
๐ฆ2
2= ๐ฆ+2
2
๐ฆ 2โ ๐ฆโ2=0
(๐ฆโ 2 ) ( ๐ฆ+1 )๐ฆ=โ1๐๐๐ 2
๐=โ ๐ฆ [โ 1,2 ]
h=( ๐ฆ+22 )โ( ๐ฆ
2
2 )h=
12
( ๐ฆ+2 โ ๐ฆ2 )
lim๐โ โ
โ 12
(๐ฆ +2 โ๐ฆ 2) โ ๐ฆ
๐ด= ๐ฆ=โ1
๐ฆ=212
( ๐ฆ+2 โ๐ฆ 2 )๐๐ฆ
๐ด=12 ( ๐ฆ
2
2+2 ๐ฆโ
๐ฆ3
3 )| 2โ1
๐ด=12 ( 22
2+2 (2 )โ 23
3 )โ 12 (โ12
2+2 (โ1 ) โ โ13
3 )๐ด=1+2โ
86
โ14+1 โ
16
๐ด=3โ2112
=1512
Multiple Regions
1) Find the points of intersections to determine the intervals.
2) Find the heights (Top minus Bottom) for each region.
3) Use the Area Addition Property.
a b c
b =
h = h =
f (x)
g (x)
x
Area : Example (x-axis - two regions):
Find the area bounded by the curve
and the x-axis.
2(1 )y x x
NOTE: The region(s) must be fully enclosed!
Area : Example ( two regions):
Find the area bounded by the curve
and . 3
1y x
NOTE: The region(s) must be fully enclosed!
1y x
Area : Example (Absolute Value):
Find the area bounded by the curve and the
x-axis on the interval x = -2 and x = 3
( ) 2 3f x x
PROBLEM 21
Velocity and Speed: Working with Absolute Value
DEFN: Speed is the Absolute Value of Velocity.
The Definite Integral of velocity is NET distance (DISPLACEMENT).
The Definite Integral of Speed is TOTAL distance. (ODOMETER).
Total Distance Traveled vs. Displacement
The velocity of a particle on the x-axis is modeled by the function, .
Find the Displacement and Total Distance Traveled of the particle on the interval, t [ 0 , 6 ]
3( ) 6x t t t
Updated:
โข 01/29/12
โข Text p 395 # 1 โ 13 odd
โข P. 396 # 15- 33 odd