Posts on integration and its applications. (The posts related to differential equations are listed here.)

Integration Itinerary (11-26-2013) Order of topics in your integration unit.

This is a long page. Scroll down or use these links to the main sections

ANTIDIFFERENTIATION

INTRODUCING THE DEFINITE INTEGRAL Riemann sums, FTC, etc.

APPLICATIONS OF INTEGRATION Area, volume, Average value of a function

ACCUMULATION

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**ANTIDIFFERENTIATION**

Antidifferentiation (11-28-2012)

Why Muss with the “+*C*”? But still don’t forget it.

Arbitrary Ranges (2-9-2014) Integrating inverse trigonometric functions.

**ANTIDIFFERENTIATION BY PARTS** This is a BC topic, or you could use it after the exam in an AB course.

Integration by Parts 1 (2-2-2013) Basics

Integration by Parts 2 (2-4-2013) The Tabular Method

Modified Tabular Integration (7-24-2013) A quicker way

Parts and More Parts (8-5-2016) Reduction formulas (Not tested on the AP Calculus exams)

Good Question 12 – Parts with a Constant (12-13-2016) How come you don’t need the “+*C*”?

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**INTRODUCING THE DEFINITE INTEGRAL**

The Old Pump (11-30-2012) An exploration and introduction to Riemann sums and integration.

Flying into Integrationland (12-3-2012) A continuation of the previous exploration.

Jobs, Jobs, Jobs (12-5-2012) Continuing the last two explorations, this time with real life data

Working Towards Riemann Sums (12-10-2012)

Riemann Sums (12-12-2012) Left, right, midpoint, and Trapezoidal Riemann sums.

Variations on a Theme – 2 (6-28-2013) Practice with Riemann sums. From the Good Question collection.

Trapezoids – Ancient and Modern (2-7-2016)

The Definition of the Definite Integral (12-14-2012) And now we’re all set for …

Foreshadowing the FTC (12-15-2014) An example shows how the FTC works.

The Fundamental Theorem of Calculus (12-17-2012) Very important and fundamental. Relating derivatives and definite integrals.

More About the FTC (12-19-2012) What the FTC really means and why it’s important.

Good Question 11 – Riemann Reversed (11-29-2016) Given a *Riemann sum* find the associated function and its domain so you can find the integral. This is the reverse of the usual problem when one finds the Riemann sum first and is **becoming a common question on the AP Calculus Exams**. There are several examples and a discussion of the concern that the answer is *never* unique, which makes it a poor question.

Properties of Integrals (12-21-2018)

Units (1-26-2018) Determining the units of a definite integral

Variations on a Theme by ETS (6-14-2013) Adapting an exam problem on area and accumulation. . From the Good Question collection.

Improper Integrals and Proper Areas (1-25-2014) A BC topic.

Math vs. the “Real World” (2-2-2018) On the convergence of improper Integrals A BC topic

Logarithms (2-6-2013) Logarithms are defined by a definite integral.

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**APPLICATIONS OF INTEGRATION**

**Area and Volume**

Area Between Curves (1-7-2013)

Under is a Long Way Down (12-7-2012) How to avoid “negative area.”

Volume of Solids with Regular Cross-sections (1-9-2013)

Volumes of Revolution (1-11-2013)

Subtract the Hole from the Whole (12-6-2016) Washer method: sometimes simplifying loses the big idea.

Does Simplifying Make Things Simpler? (4-4-2013)

Why You Never Need Cylindrical Shells (1-14-2013) But they are easier. (This topic is not tested on the AP Calculus exams.

Painting a Point (2-4-2013) Paint often and the paint accumulates.

**VISUALIZING SOLID FIGURES**

Visualizing Solid Figures 1 (11-13-2014) Physical models

Visualizing Solid Figures 2 (11-16-2014) Winplot Instructions – solids with regular cross-sections.

Visualizing Solid Figures 3 (11-19-2014) Winplot – Disk and Washer method

Visualizing Solid Figures 4 (11-22-2014) Winplot – Cylindrical Sells

Visualizing Solid Figures 5 (11-25-2014) Winplot and also see Challenge (3-29-2013) and Challenge Answer (4-8-20130 A question on volume by washers and by cylindrical shells shows how accumulation can be accomplished different ways.

**Review Notes:** Type 4 Questions: Area and Volume (3-16-2018) Review Notes

**AVERAGE VALUE OF A FUNCTION**

Half-full or Half-empty (1-16-2015) A thought experiment on average value of a function.

Average Value of a Function (1-16-2013)

What’s a Mean Old Average Anyway? (4-29-2014) Helping students understand the difference between the *average rate of change of a function,* the a*verage value of a function*, and the *Mean Value theorem*

Most Triangles Are Obtuse! (1-18-2013) An application of the average value of a function.

Average Value Activity Discovering the average value of a function

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**ACCUMULATION**

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

Accumulation: Need an Amount? (1-21-2013) An important and always tested application.

AP Accumulation Questions (1-23-2013) Two good questions for teaching and learning accumulation.

Graphing with Accumulation 1 (1-25-2013) 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 (1-28-2013) Everything you need to know about the graph of a function given its derivative can be found using integration techniques*. Concavity*.

Painting a Point (2-4-2013) Paint often and the paint accumulates.

Good Question 6: 2000 AB 4 (8-25-2015) Accumulation

Good Question 8 – or not? (1-5-2016) Accumulation

Density (1-10-2017)

Accumulation and Differential Equations (2-1-2013) Solving differential equations without the “+*C*“

**Review Notes:** Type 1 Questions: Rate and Accumulation (3-6-2018) Review Notes