Unit 6 – Integration and Accumulation

This unit develops the beginnings of integration and accumulation. Riemann sums lead to the definition of the definite integral, the Fundamental Theorem of Calculus, and finding antiderivatives. Density, while not listed on the CED, is an application of integration seen sometimes on tests.

Getting Started

Integration Itinerary – A discussion of the order in which the integration topics may be taught.

A Calculus Journey – A look ahead to presenting the theory on integration.

Antidifferentiation – The basics antidifferentiation: pattern recognition.

The Old Pump and Flying into Integrationland – to similar exploratory exercises to introduce integration.

Jobs, Jobs, Jobs = Another exploratory exercise: real world example of integration.

Under is a Ling Way Down – start by using the correct terminology.

Riemann Sums

Working Towards Riemann Sums – partitioning an interval and approximating areas.

Riemann Sums – Left-, right-, midpoint-, and trapezoidal sums.

Variations on a Theme – 2 – Putting Riemann sums in order.

Good Question 4: 2008 AB 10 – Comparing the sizes of the Riemann sums.

Trapezoids – Ancient and Modern – the ancient Babylonians used Trapezoidal approximations. And a reasonable question on Riemann sums.

The Definite Integral

The Definition of the Definite Integral – A very important definition.

Good Question 11 – Riemann Reversed – Given an integral find its Reimann sum – a common AP Exam questions (but it has problems).

The Fundamental Theorem of Calculus

Foreshadowing the FTC – An example evaluating a definite without the FTC, but with the same idea.

The Fundamental Theorem of Calculus – and what it says.

The Definite Integral and the FTC – What to do after you present the FTC to show how useful it will be.

More on the FTC – the derivative of the integral is the integrand. 

Y the FTC? – Two things to do as soon as you’ve shown students the FTC

Antiderivatives & Properties of Integrals

Properties of Integrals – continuing the exercise from the previous post.

Definite integrals – Exam Considerations – (7) calculator integration

Integration by Parts 1 – Why it works

Integration by Parts 2 – The Tabular method and reduction formulas.

Modified Tabular Integration – A simpler method for antiderivatives that require two or more integrations by parts.

Parts and More Parts – More on tabular integration.

Good Question 12 – Parts with a Constant? – How come you don’t need a constant of integration when doing integration by parts?

Good Question 13 – Four instructive ways to do a simple review problem on integration.

Graphing Integrals – (6) Graphing integrals on a calculator without integrating.

Variation on a Theme by ETS – A simple problem with lots of potential: 2008 AB calculus exam, question 9.

Why Muss with the “+C”? – Another way of dealing with the initial condition while integrating.

Improper Integrals and Proper Areas – On the range of the inverse tangent.

Math vs. the “Real World” On adding improper integrals.

Accumulation

Good Question 6: 2000 AB 4 – One of my favorite questions is great for teaching accumulation and in-out questions.

Good Question 7 – 2009 AB 3 – The Mighty Cable Company: an in-out question and good practice in reading and translating an unfamiliar situation into mathematics.

Density

Density Functions – Not quite in the CED, but maybe on the exam.

Good Question 15: 2018 BC 2(a) – A density question and a reading question.

Good Question 16: 2018 BC 2(b) – Good question 15 continued.