EE101 Sen Lnt 001 Area May11
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Transcript of EE101 Sen Lnt 001 Area May11
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IntegrationIntroduction
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The Definite Integral as a Limit of
Reimann Sums
Letfbe a function defined on a closed interval [a,b]. For any
partition P of [a,b], let the numbers ckbe chosen arbitrarily in
the subintervals [xk-1, xk].
If there exists a number I such that
Then,fis integrable on [a,b] and I is the definite integral offover [a,b].
Ixcfn
k
kk
P
10||||
)(lim
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The Definite Integral
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0bb],[0,
intervalover theunderareathefindand
Compute0
xyA
xxb
Example 1
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Find the average value of 3,0on1)(
2 xxf
Example 2
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Solution:
3
0
21
03
1dxxavg
033
33
3
1
33
1 3
0
3
x
x
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Example 3
Calculate the area bounded by
the x-axis and the parabola
26 xxy
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Solution:
2
3
2
3
322
326)6(
xx
xdxxx
3
27
2
918
3
8212
6
520
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Example 4
212)(
ofgraphtheand
axis-between x
regiontheof
areatheFind
23
xxxxxf
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12
5
34
)2(
:
0
1
234
0
1
23
x
xx
dxxxx
Solution
3
834
)2(
2
0
234
2
0
23
x
xx
dxxxx
2units12
37
3
8
12
5areaenclosedTotal
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b
a
dxxgxfArea )]()([
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Example 5
Find the area
enclosed by the
parabolay=2-x2
and the liney=-x.
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Find the solution by 3 steps:
1. Sketch the curves and shade the required
area.
2. Find the limits of integration by solving
simultaneous equations.
3. Calculate the area.
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Solution:
2band-1aarenintegratioofLimits
-2y2,When x1y-1,When x
2,1
0)2)(1(02
2
and2Solving
2
2
2
xx
xx
xx
xx
xyxy
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2
9
3
1
2
1
23
8
2
4
4
322
)2(
)]()2[(
)]()([
2
1
32
2
1
2
2
1
2
xxx
dxxx
dxxx
dxxgxfA
b
a
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Example 6
6.
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Solution:
Area A
Limits of integration are a=0, b=2.
Area B
Solving simultaneously 2and xyxy
4,1
0)4)(1(
045
44)2(
2
22
xx
xx
xx
xxxx
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310
)2(areaTotal
4
2
2
0
dxxxdxx
2)-(x-)()(:42For
)()(:20For
4limithandRight
xxgxfx
xxgxfx
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Solution:
Solving simultaneous equation of
gives roots y=-1, y=2.
Limits a=0,b=2.
2and2 yxyx
3
10
)2(
)]()([
2
0
2
dyyy
dyygyfAb
a
