copyright © 2015, 2012, and 2009 pearson education, inc. 1 section 6.4 fundamental theorem of...

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1

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