section 5.4 theorems about definite integrals
DESCRIPTION
Section 5.4 Theorems About Definite Integrals. Properties of Limits of Integration If a , b , and c are any numbers and f is a continuous function, then. Properties of Sums and Constant Multiples of the Integrand Let f and g be continuous functions and let c be a constant, then. - PowerPoint PPT PresentationTRANSCRIPT
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Section 5.4Theorems About Definite Integrals
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Properties of Limits of Integration
• If a, b, and c are any numbers and f is a continuous function, then
b
a
a
bdxxfdxxf )()(
b
a
b
c
c
adxxfdxxfdxxf )()()(
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Properties of Sums and Constant Multiples of the Integrand
• Let f and g be continuous functions and let c be a constant, then
b
a
b
a
b
a
b
a
b
a
dxxfcdxxcf
dxxgdxxfdxxgxf
)()()2
)()())()(()1
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Example• Given that
find the following:
cbadxxfdxxfc
a
b
a and,10)(,5)(
c
bdxxf )(
b
cdxxf )(
c
adxxf )(3
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Using Symmetry to Evaluate Integrals
• An EVEN function is symmetric about the y-axis
• An ODD function is symmetric about the originIf f is EVEN, then
If f is ODD, then
aa
adxxfdxxf
0)(2)(
0)( a
adxxf
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EXAMPLE
Given that
Find
2sin0
xdx
dxxb
xdxa
|sin|__.
sin1__.
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Comparison of Definite Integrals• Let f and g be continuous functions
)()()(then
,for)(If1)
abMdxxfabm
bxaMxfmb
a
b
a
b
adxxgdxxf
thenbxaforxgxf
)()(
,)()(If)2
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Example
• Explain why
03)cos( dxx
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The Area Between Two Curves
• If the graph of f(x) lies above the graph of g(x) on [a,b], then
b
adxxgxf ))()((
Area between f and g on [a,b]
Let’s see why this works!
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Find the exact value of the area between the graphs of
y = e x + 1 and y = xfor 0 ≤ x ≤ 2
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This is the graph of y = e x + 1 What does the integral from 0 to 2 give us?
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Now let’s add in the graph of y = x
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Now the integral of x from 0 to 2 will give us the area under x
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So if we take the area under e x + 1 and subtract out the area under x, we get the area between the 2 curves
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So we find the exact value of the area between the graphs of
y = e x + 1 and y = xfor 0 ≤ x ≤ 2 with the integral
2 2 2
0 0 0( 1) ( 1 )x xe dx xdx e x dx
Notice that it is the function that was on top minus the function that was on bottom
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Find the exact value of the area between the graphs of
y = x + 1 and y = 7 - x for 0 ≤ x ≤ 4
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Let’s shade in the area we are looking for
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Notice that these graphs switch top and bottom at their intersectionThus we must split of the integral at the intersection point and switch the order
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So to find the exact value of the area between the graphs of
y = x + 1 and y = 7 - x for 0 ≤ x ≤ 4 we can use the following integral
3 4
0 3
3 4
0 3
(7 ) ( 1) ( 1) (7 )
(6 2 ) (2 6)
x x dx x x dx
x dx x dx
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Find the exact value of the area enclosed by the graphs of
y = x2 and y = 2 - x2
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Let’s shade in the area we are looking for
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In this case we weren’t given limits of integrationSince they enclose an area, we use their intersection points for the limits
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So to find the exact value of the area enclosed by the graphs of
y = x2 and y = 2 - x2 we can use the following integral
1 12 2 2
1 1(2 ) ( ) (2 2 )x x dx x dx