# Unit 8 Applications of Integration

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ENDURING UNDERSTANDING

CHA-4 Definite integrals allow us to solve problems involving the accumulation of change over an interval.

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 Topic Name Essential Knowledge 8.1 Finding the Average Value of a Function on an Interval LEARNING OBJECTIVE CHA-4.B Determine the average value of a function using definite integrals. CHA-4.B.1 The average value of a continuous function f over an interval [a, b] is $\displaystyle \frac{1}{{b-a}}\int_{a}^{b}{{f\left( x \right)dx}}$

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Half-full or Half-empty  A thought experiment on average value of a function.

Average Value of a Function

What’s a Mean Old Average Anyway?  Helping students understand the difference between the average rate of change of a  function, the average value of a function, and the Mean Value theorem

Most Triangles Are Obtuse! Proved by a tenth grade BC student. An application of the average value of a function.

Average Value Activity Discovering the average value of a function

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 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals LEARNING OBJECTIVE CHA-4.C Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion. CHA-4.C.1 For a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle’s displacement over the interval of time, and the definite integral of speed represents the particle’s total distance traveled over the interval of time.

Blog Posts

The Ubiquitous Particle Motion Problem  – a PowerPoint Presentation and its Handout

Motion Matching  A quick quiz

Speed

Speed Activity An exploration on Speed

A Note on Speed An analytic approach

Brian Leonard’s Particle Motion Game Velocity Game  and

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 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts LEARNING OBJECTIVES CHA-4.D Interpret the meaning of a definite integral in accumulation problems. CHA-4.E Determine net change using definite integrals in applied contexts. CHA-4.D.1 A function defined as an integral represents an accumulation of a rate of change. CHA-4.D.2 The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval. CHA-4.E.1 The definite integral can be used to express information about accumulation and net change in many applied contexts.

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Accumulation An introductory activity to explore accumulation and the relationship between an accumulation and derivatives

Accumulation: Need an Amount?   An important and always tested application.

AP Accumulation Questions Two good questions for teaching and learning accumulation.

Graphing with Accumulation 1  Everything you need to know about the graph of a function given its derivative can be found using integration techniques. Increasing and decreasing.

Graphing with Accumulation 2 Everything you need to know about the graph of a function given its derivative can be found using integration techniques. Concavity.

Painting a Point Paint often and the paint accumulates.

Good Question 6: 2000 AB 4 Accumulation

Good Question 8 – or not? Accumulation

Density

Accumulation and Differential Equations  Solving differential equations without the “+C

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ENDURING UNDERSTANDING

CHA-5 Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

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 8.4 Finding the Area Between Curves Expressed as Functions of x LEARNING OBJECTIVE CHA-5.A Calculate areas in the plane using the definite integral. CHA-5.A.1 Areas of regions in the plane can be calculated with definite integrals.

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Area Between Curves

Under is a Long Way Down How to avoid “negative area.”

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 8.5 Finding the Area Between Curves Expressed as Functions of y LEARNING OBJECTIVE CHA-5.A Calculate areas in the plane using the definite integral. CHA-5.A.2 Areas of regions in the plane can be calculated using functions of either x or y.

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Area Between Curves

Under is a Long Way Down How to avoid “negative area.”

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 8.6 Finding the Area Between Curves That Intersect at More Than Two Points LEARNING OBJECTIVE CHA-5.A Calculate areas in the plane using the definite integral. CHA-5.A.3 Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.

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Area Between Curves

Under is a Long Way Down How to avoid “negative area.”

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 8.7 Volumes with Cross Sections: Squares and Rectangles LEARNING OBJECTIVE CHA-5.B Calculate volumes of solids with known cross sections using definite integrals. CHA-5.B.1 Volumes of solids with square and rectangular cross sections can be found using definite integrals and the area formulas for these shapes.

Blog Posts

Subtract the Hole from the Whole  Washer method: sometimes simplifying loses the big idea.

Does Simplifying Make Things Simpler?

Why You Never Need Cylindrical Shells But they are easier. (This topic is not tested on the AP Calculus exams.

Painting a Point Paint often and the paint accumulates.

VISUALIZING SOLID FIGURES

Visualizing Solid Figures 1  Physical models

Visualizing Solid Figures 2  Winplot Instructions – solids with regular cross-sections.

Visualizing Solid Figures 3 Winplot – Disk and Washer method

Visualizing Solid Figures 4  Winplot – Cylindrical Sells

Visualizing Solid Figures 5  Winplot and also see Challenge  and Challenge Answer A question on volume by washers and by cylindrical shells shows how accumulation can be accomplished different ways.

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 8.8 Volumes with Cross Sections: Triangles and Semicircles LEARNING OBJECTIVE CHA-5.B Calculate volumes of solids with known cross sections using definite integrals. CHA-5.B.2 Volumes of solids with triangular cross sections can be found using definite integrals and the area formulas for these shapes. CHA-5.B.3 Volumes of solids with semicircular and other geometrically defined cross sections can be found using definite integrals and the area formulas for these shapes.

Blog Posts

Subtract the Hole from the Whole  Washer method: sometimes simplifying loses the big idea.

Does Simplifying Make Things Simpler?

Why You Never Need Cylindrical Shells But they are easier. (This topic is not tested on the AP Calculus exams.

Painting a Point Paint often and the paint accumulates.

VISUALIZING SOLID FIGURES

Visualizing Solid Figures 1  Physical models

Visualizing Solid Figures 2  Winplot Instructions – solids with regular cross-sections.

Visualizing Solid Figures 3 Winplot – Disk and Washer method

Visualizing Solid Figures 4  Winplot – Cylindrical Sells

Visualizing Solid Figures 5  Winplot and also see Challenge  and Challenge Answer A question on volume by washers and by cylindrical shells shows how accumulation can be accomplished different ways.

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 8.9 Volume with Disk Method: Revolving Around the x– or y-Axis LEARNING OBJECTIVE CHA-5.C Calculate volumes of solids of revolution using definite integrals. CHA-5.C.1 Volumes of solids of revolution around the  x- or y-axis may be found by using definite integrals with the disc method.

Blog Posts

Subtract the Hole from the Whole  Washer method: sometimes simplifying loses the big idea.

Does Simplifying Make Things Simpler?

Why You Never Need Cylindrical Shells But they are easier. (This topic is not tested on the AP Calculus exams.

Painting a Point Paint often and the paint accumulates.

VISUALIZING SOLID FIGURES

Visualizing Solid Figures 1  Physical models

Visualizing Solid Figures 2  Winplot Instructions – solids with regular cross-sections.

Visualizing Solid Figures 3 Winplot – Disk and Washer method

Visualizing Solid Figures 4  Winplot – Cylindrical Sells

Visualizing Solid Figures 5  Winplot and also see Challenge  and Challenge Answer A question on volume by washers and by cylindrical shells shows how accumulation can be accomplished different ways.

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 8.10 Volume with Disk Method: Revolving Around Other Axes LEARNING OBJECTIVE CHA-5.C Calculate volumes of solids of revolution using definite integrals. CHA-5.C.2 Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.

Blog Posts

Subtract the Hole from the Whole  Washer method: sometimes simplifying loses the big idea.

Does Simplifying Make Things Simpler?

Why You Never Need Cylindrical Shells But they are easier. (This topic is not tested on the AP Calculus exams.

Painting a Point Paint often and the paint accumulates.

VISUALIZING SOLID FIGURES

Visualizing Solid Figures 1  Physical models

Visualizing Solid Figures 2  Winplot Instructions – solids with regular cross-sections.

Visualizing Solid Figures 3 Winplot – Disk and Washer method

Visualizing Solid Figures 4  Winplot – Cylindrical Sells

Visualizing Solid Figures 5  Winplot and also see Challenge  and Challenge Answer A question on volume by washers and by cylindrical shells shows how accumulation can be accomplished different ways.

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 8.11 Volume with Washer Method: Revolving Around the x– or y-Axis LEARNING OBJECTIVE CHA-5.C Calculate volumes of solids of revolution using definite integrals. CHA-5.C.3 Volumes of solids of revolution around the x- or y-axis whose cross sections are ring shaped may be found using definite integrals with the washer method.

Blog Posts

Subtract the Hole from the Whole  Washer method: sometimes simplifying loses the big idea.

Does Simplifying Make Things Simpler?

Why You Never Need Cylindrical Shells But they are easier. (This topic is not tested on the AP Calculus exams.

Painting a Point Paint often and the paint accumulates.

VISUALIZING SOLID FIGURES

Visualizing Solid Figures 1  Physical models

Visualizing Solid Figures 2  Winplot Instructions – solids with regular cross-sections.

Visualizing Solid Figures 3 Winplot – Disk and Washer method

Visualizing Solid Figures 4  Winplot – Cylindrical Sells

Visualizing Solid Figures 5  Winplot and also see Challenge  and Challenge Answer A question on volume by washers and by cylindrical shells shows how accumulation can be accomplished different ways.

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 8.12 Volume with Washer Method: Revolving Around Other Axes LEARNING OBJECTIVE CHA-5.C Calculate volumes of solids of revolution using definite integrals. CHA-5.C.4 Volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped may be found using definite integrals with the washer method.

Blog Posts

Subtract the Hole from the Whole  Washer method: sometimes simplifying loses the big idea.

Does Simplifying Make Things Simpler?

Why You Never Need Cylindrical Shells But they are easier. (This topic is not tested on the AP Calculus exams.

Painting a Point Paint often and the paint accumulates.

VISUALIZING SOLID FIGURES

Visualizing Solid Figures 1  Physical models

Visualizing Solid Figures 2  Winplot Instructions – solids with regular cross-sections.

Visualizing Solid Figures 3 Winplot – Disk and Washer method

Visualizing Solid Figures 4  Winplot – Cylindrical Sells

Visualizing Solid Figures 5  Winplot and also see Challenge  and Challenge Answer A question on volume by washers and by cylindrical shells shows how accumulation can be accomplished different ways.

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ENDURING UNDERSTANDING

CHA-6 Definite integrals allow us to solve problems involving the accumulation of change in length over an interval.

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 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled BC ONLY LEARNING OBJECTIVE CHA-6.A Determine the length of a curve in the plane defined by a function, using a definite integral. BC ONLY CHA-6.A.1 The length of a planar curve defined by a function can be calculated using a definite integral. BC ONLY

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