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Section 6.4
Fundamental Theorem of Calculus
Applications of Derivatives
Chapter 6
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Quick Review
3
3
2 2
Find / .
1. sin
2. sin
3. ln 3 ln 7
4. sin cos
5. 3
6. cos
7. sin and 2
8. / 2
x
dy dx
y x
y x
y
y x x
y
xy
xy t x t
dx dy x
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Quick Review Solutions
3
3
2 2
2 3
2
2
Find / .
1. sin
2. sin
3. ln 3 ln 7
4. s
/ 3 cos
/ 3 sin cos
/ 0
in / 0
/ 3 l
cos
5. 3
6. cos
7. sin
n
and
3
cos sin/
c s
o
xx
dy dx
y x
y x
y
y x x
y
dy dx x x
dy dx x x
dy dx
dy dx
dy dx
x x xxy
xdy
x
t
dx
y x
2
8. / 2
cos/
21
/2
t
dx d
tdy dx
dy dxx
y x
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What you’ll learn about The Antiderivative Part of the Fundamental Theorem of
Calculus Use of definite integrals to define new functions (accumulator
functions) The Evaluation Part of the Fundamental Theorem of Calculus Evaluation of definite integrals using antiderivatives
… and whyThe Fundamental Theorem of Calculus is a Triumph Of Mathematical Discovery and the key to solving many problems.
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The Fundamental Theorem of Calculus
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The Fundamental Theorem of Calculus
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Example Applying the Fundamental Theorem
Find sin .xd
tdtdx
sin sinxd
tdt xdx
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Example The Fundamental Theorem with the Chain Rule
2
1Find / if sin .
xdy dx y tdt
2
1sin
xy tdt
2
1sin and .u
y tdt u x Apply the chain rule:
dy dy du
dx du dx
1sinud du
tdtdu dx
sindu
udx
sin 2u x 22 sinx x
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Example Variable Lower Limits of Integration
5
Find if sin .x
dyy t tdt
dx
5
5sin sin
x
x
d dt tdt t tdt
dx dx
5sin
xdt tdt
dx
sinx x
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The Fundamental Theorem of Calculus, Part 2
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The Fundamental Theorem of Calculus, Part 2
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Example Evaluating an Integral
3 2
1Evaluate 3 1 using an antiderivative.x dx
3 32 3
113 1x dx x x
333 3 1 1
32
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How to Find Total Area Analytically
To find the area between the graph of ( ) and the -axis over the
interval [ , ] analytically,
1. partition [ , ] with the zeros of ,
2. integrate over each subinterval,
3. add the absolute values
y f x x
a b
a b f
f
of the integrals.
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How to Find Total Area Numerically
To find the area between the graph of ( ) and the -axis over the
interval [ , ] numerically, evaluate
NINT(| ( ) |, , , )
y f x x
a b
f x x a b