# Unit 8 – Applications of Integration

The applications tested on the AP Calculus exams include average value, accumulation, finding area, finding volumes

Units – How to determine the units of a derivative and a definite integral.

Average Value

Average Value of a Function – An important application of integration

Most Triangles Are Obtuse! – In fact, 81.865% of all triangles. An average value problem solved by a tenth grader.

What’s a Mean Old Average Anyway? – Helping your students with the average rate of change, the average value of a function, and the average value of a function.

Half-full and Half-empty – Visualizing the average value of a function.

Extreme Average – Exploring the average extreme value

Accumulation

Accumulation: Need an Amount? – Look around for a rate to integrate.

AP Accumulation Questions – A discussion of two of my favorite AP Exam questions about accumulation.

Graphing with Accumulation 1 – Using the graph of a derivative and accumulation to find extreme values without mentioning the derivative.

Graphing and Accumulation 2 – Using the graph of a derivative and accumulation to determine concavity without mentioning the derivative.

Accumulation and Differential Equations – Solving simple differential equations

Painting a Point – accumulating paint.

Area and Volume

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

Area between Curves – The best way to think of it even if one curve is the x-axis.

Volumes of Solids with Regular Cross-sections – An important application

Volumes of Revolution – A special case of the volume problem.

Subtract the Hole from the Whole – Easier than R2 – r2

Why You Never Need Cylindrical Shells – but often they are the easiest way.

Challenge and its Solution – Seeing the difference between two methods of doing the same volume problem.

Does Simplifying Make Things Simpler? – Yes, no, and maybe.

Who’d a thunk it? – An exploration into some strange area ratios.

Lin McMullin’s Theorem – tripping over the Golden Ratio. And a related result More Gold

Visualizing Solid Figures

…but what does it look like? – Ways to see solid figures

Visualizing Solid Figures 1 – Physical models

Visualizing Solid Figures 2 = Using Winplot*

Visualizing Solid Figures 3 – The Washer Method, Winplot*

Visualizing Solid Figures 4 – Cylindrical Shells, Winplot*

Visualizing Solid Figures 5 – Comparing Washers and Shells

Other

Logarithms – Defining the logarithm function as the inverse of the exponential function

*Winplot is an excellent graphing program which has gone out of fashion. I still use it. The program may be downloaded for free HERE. There are instructions on the internet and in my posts that refer to it.