The Fundamental Theorem of Calculus Inverse Operations.
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Transcript of The Fundamental Theorem of Calculus Inverse Operations.
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The Fundamental Theorem of Calculus
Inverse Operations
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Fundamental Theorem of Calculus
Discovered independently by Gottfried Liebnitz and Isaac Newton
Informally states that differentiation and definite integration are inverse operations.
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Fundamental Theorem of Calculus
If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b], then
( ) ( ) ( )b
af x dx F b F a
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Guidelines for Using the Fundamental Theorem of Calculus
1. Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum.
2. When applying the Fundamental Theorem of Calculus, the following notation is used
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Guidelines
( ) ( )] ( ) ( )b b
aaf x dx F x F b F a
It is not necessary to include a constant of integration C in the antiderivative because they cancel out when you subtract.
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Evaluating a Definite Integral
7 7 722 2
Evaluate each definite integral.
3dv 3 3 ] 21 6 15dv v
1
22
1( )u du
u
21 2 1 1
22
2 2
]2
1 21 1
2 1 2 2
1 32
2 2
uu u dx u
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Evaluate the Definite Integral
24
20
1 sin
cos
2
4 420 0
cos1
cosd d
40] 0
4 4
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Evaluate the Definite Integral
24
0sec x dx
4
0tan tan tan 0 1 0 1
4x
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Definite Integral Involving Absolute Value Evaluate
2
02 1
The absolute value function has to be broken up
into its two parts:
12 1
22 11
2 12
x dx
x if xx
x if x
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Definite Integral Involving Absolute Value
12
21
02
122 221
02
2 22
Now evaluate each part separately
2 1 (2 1)
1 1 1 10 2 2
2 2 2 2
1 1 52
4 4 2
x dx x dx
x x x x
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Using the Fundamental Theorem to Find Area
Find the area of the region bounded by the graph of y = 2x3 – 3x + 2, the x-axis, and the vertical lines x = 0 and
x = 2
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Using the Fundamental Theorem to Find Area
2 3
0
24 2
0
2 3 2
2 32
4 2
8 6 4 0 0 0
6
Area x x dx
x xx
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The Mean Value Theorem for Integrals
If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that
( ) ( )( ).b
af x dx f c b a
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Average Value of a Function
This is just another way to write the Mean Value Theorem (mean = average in mathematics)
If f is integrable on the closed interval
[a,b], then the average value of f on the interval is
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Average Value of a Function
( )b
af x dx
cb a
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Finding the Average Value of a Function
Find the average value of f(x) = sin x on the interval [0, ]
0 0sin cos
01 1 2
x dx xc
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Force
The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of sec x, where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is
[0, /3] and F(0) = 500.
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Force
(a) Find F as a function of x.
F(x) = 500 sec2 x
(b) Find the average force exerted by the press over the interval [0, /3]
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Force
32
0
30
500 sec
03
500 tan
3
500 tan tan 03
3
x dx
F
x
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Force
500 3 0
3
500 3 1500 3827
3
Newtons
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Second Fundamental Theorem of Calculus
If f is continuous on an open interval I containing a, then, for every x in the interval,
( ) ( )x
a
df t dt f x
dx
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Using the Second Fundamental Theorem of Calculus
Evaluate
2
0( ) ( 1)
xdF x t t dt
dx
2 3( 1)x x x x
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Second Fundamental Theorem of Calculus
Find F’(x) of
2
32
1( )
xF x dt
t
6 5
1 22x
x x