MAT 1235 Calculus II Section 5.1 Area Between Curves .
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MAT 1235 Calculus II Section 5.1 Area Between Curves http://myhome.spu.edu/lauw
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Transcript of MAT 1235 Calculus II Section 5.1 Area Between Curves .
MAT 1235Calculus II
Section 5.1
Area Between Curves
http://myhome.spu.edu/lauw
Homework
WebAssign HW 5.1
Preview
Want to compute the area of the region between two curves.
Two types of basic regions• Bounded above and below• Bounded on the left and on the right
Composite region• Break the region up into smaller basic regions
The Area Problem
The Area Problem
( )
( ) 0 on [ , ]
y f x
f x a b
Abstract World
1
lim ( )n
ini
A f x x
The Energy Problem
( )y f x
( )f x
Real World
Two Important Themes
The formula for computations The process of finding the formula
• More important in a long run• The processes of finding integral based
formula (geometric properties, physical properties) are similar.• Volume, Surface area, Arc length, Work Done,
Center of Mass, …
Recall: Definition
The Definite Integral of from to
n
ii
n
b
axxhdxxh
1
)(lim)(
nabx /)(
In other words…
Any limit of the form :
can be interpreted as
b
a
n
ii
ndxxhxxh )()(lim
1
n
ii
nxxh
1
)(lim
Type I Region: Region Bounded Above/Below
b
)(xfy
)(xgy
a
Type I Region: Region Bounded Above/Below
ith subinterval
ix
sample point
)()( ii xgxf
a b
)(xfy
)(xgy
Type I Region: Region Bounded Above/Below
Area of each rectangle:
b
a
n
iii
n
dxxgxf
xxgxf
)()(
)()(lim1
xxgxf ii )()(
Area of the region:
b
a
n
ii
ndxxhxxh )()(lim
1
Type I Region: Region Bounded Above/Below
Area of each rectangle:
b
a
n
iii
n
dxxgxf
xxgxf
)()(
)()(lim1
xxgxf ii )()(
Area of the region:
Key: To know which curve is above/below
Example 1
Find the area of the region enclosed by
and2xy 21 xy
Example 1Step 1: Sketch the graphs
2xy 21 xy
Example 1Step 1: Sketch the graphs
21 xy
2xy
? limit lower ? limit upper
2xy 21 xy
2 2
3
Type II Region: Region Bounded Left/Right
d
)(yfx
)(ygx
c
d
c
dyygyf )()(Area
Example 2
Find the area of the region enclosed by
and12 yx 1yx
Example 2 12 yx 1yx
>with(plots):
>implicitplot([x=y^2-1,x=y+1],x=-2..4,y=-2..4);
Example 2 12 yx 1yx
12 yx1yx
9
2
Summary
1. Sketch the curves• Determine the type of the region
(above/below, left/right)• Determine the nature of each curve
2. Find the upper/lower limits
3. Compute the integrals
