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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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Page 1: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

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

Section 6.4

Fundamental Theorem of Calculus

Applications of Derivatives

Chapter 6

Page 2: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 2

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

Page 3: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 3

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

Page 4: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 4

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.

Page 5: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 5

The Fundamental Theorem of Calculus

Page 6: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 6

The Fundamental Theorem of Calculus

Page 7: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 7

Example Applying the Fundamental Theorem

Find sin .xd

tdtdx

sin sinxd

tdt xdx

Page 8: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 8

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

Page 9: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 9

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

Page 10: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 10

The Fundamental Theorem of Calculus, Part 2

Page 11: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 11

The Fundamental Theorem of Calculus, Part 2

Page 12: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 12

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

Page 13: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 13

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.

Page 14: Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.4 Fundamental Theorem of Calculus Applications of Derivatives Chapter 6

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 14

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


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