Integral Calculus

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

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



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”?



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.



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 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


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 average 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



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