American College of Computer & Information Sciences
Microscopy Optimization
ENGINEERIN 2012

Spring 2013
Applications of the Denite Integral
Volumes of Solids of Revolution
Mathematics 53
Institute of Mathematics  UP Diliman
7 September 2012
Mathematics 53 (Part 4)
Arc Length and Volumes by Slicing
7 September 2012
1 / 64
For today
1
Solids of Revolution: D
American College of Computer & Information Sciences
Microscopy Optimization
ENGINEERIN 2012

Spring 2013
Applications of the Denite Integral
Arc Length, Volumes by Slicing,
Solids of Revolution: Disk or Washer Method
Mathematics 53
Institute of Mathematics  UP Diliman
5 September 2012
Mathematics 53 (Part 4)
Arc Length and Volumes by Slicing
5 September 201
American College of Computer & Information Sciences
Microscopy Optimization
ENGINEERIN 2012

Spring 2013
Applications of the Denite Integral
The Area of a Plane Region
Mathematics 53
Institute of Mathematics  UP Diliman
1 September 2012
Mathematics 53 (Part 4)
Area of a Plane Region
1 September 2012
1 / 59
Review of Parabolas and Circles
Parabolas opening u
American College of Computer & Information Sciences
Microscopy Optimization
ENGINEERIN 2012

Spring 2013
The Denite Integral and
the Fundamental Theorem of Calculus
Mathematics 53
Part 4
Institute of Mathematics
Mathematics 53 (Part 4)
The Denite Integral
Institute of Mathematics
1 / 51
Outline
1
The Denite Integral
2
MeanValue Theorem for Integrals
3
The F
American College of Computer & Information Sciences
Microscopy Optimization
ENGINEERIN 2012

Spring 2013
Integration by Substitution
Introduction to the Area of a Plane Region
Mathematics 53
Institute of Mathematics  UP Diliman
24 August 2012
Math 53 (Part 4)
Substitution and Area
24 August 2012
1 / 44
Outline
1
Integration by Substitution
2
Introduction th
American College of Computer & Information Sciences
Microscopy Optimization
ENGINEERIN 2012

Spring 2013
Antidierentiation and the Indenite Integral
Part 4: Integration and Its Applications
Mathematics 53
Institute of Mathematics  UP Diliman
22 August 2012
Math 53 (Part 4)
The Indenite Integral
22 August 2012
1 / 32
Outline
1
Antiderivatives or Indenite Int
American College of Computer & Information Sciences
Polymorphic Trigonometry
ENGINEERIN 2012

Spring 2013
Graphing a Function Using its Derivatives
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.3
Recall:
The graph of f .
is increasing if f (x) > 0
Recall:
The graph of f .
is increasing if f (x) > 0
is decreasing if f (x) < 0
Recall:
The graph
American College of Computer & Information Sciences
Polymorphic Trigonometry
ENGINEERIN 2012

Spring 2013
Graphing a Function Using its Derivatives
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.3
Recall:
The graph of f .
is increasing if f (x) > 0
Recall:
The graph of f .
is increasing if f (x) > 0
is decreasing if f (x) < 0
Recall:
The graph
American College of Computer & Information Sciences
Polymorphic Trigonometry
ENGINEERIN 2012

Spring 2013
Graphing a Function Using its Derivatives
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.3
Recall:
The graph of f .
is increasing if f (x) > 0
Recall:
The graph of f .
is increasing if f (x) > 0
is decreasing if f (x) < 0
Recall:
The graph
American College of Computer & Information Sciences
Polymorphic Trigonometry
ENGINEERIN 2012

Spring 2013
The Mean Value Theorem and Relative Extrema
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.1
For today
1
The Mean Value Theorem
2
Relative Extrema of a Function
3
Increasing/Decreasing Functions
3
The First Derivative Test For Relative Ext
American College of Computer & Information Sciences
Polymorphic Trigonometry
ENGINEERIN 2012

Spring 2013
The Mean Value Theorem and Relative Extrema
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.1
For today
1
The Mean Value Theorem
2
Relative Extrema of a Function
3
Increasing/Decreasing Functions
3
The First Derivative Test For Relative Ext