# Applications of Integration, part 3: Accumulation

Integration, at its basic level, is addition. A definite integral is a sum (a Riemann sum). When you add things you get an amount of whatever you are adding: you accumulate. Here are some previous posts on this important idea that often shows up on the AP Calculus exams (usually the first free-response question!) Accumulation:…

# Rate and Accumulation Questions (Type 1)

The Free-response Questions The free-response questions fall into 10 general categories or types. The multiple-choice questions fall largely into the same categories plus some straight-forward questions asking students to find limits, derivatives, and integrals. Often two or more type are combined into one question. The types are the following. Rate and Accumulation Linear motion Graph Analysis…

# Good Question 7 – 2009 AB 3

Another in my occasional series on Good Questions to teach from. This is the Mighty Cable Company question from the 2009 AB Calculus exam, number 3 This question presented students with a different situation than had been seen before. It is a pretty standard “in-out” question, except that what was going in and out was money.…

# The Rate/Accumulation Question

AP Type Questions 1 These questions are often in context with a lot of words describing a situation in which some things are changing. There are usually two rates given acting in opposite ways. Students are asked about the change that the rates produce over some time interval either separately or together. The rates are…

# Painting a Point

Accumulation 7: An application (of paint) Suppose you started with a point, the origin to be specific, and painted it. You put on layers and layers of paint until your point grows to a sphere with radius r. Let’s stop and admire your work part way through the job; at this point the radius is…

# Stamp Out Slope-intercept Form!

Accumulation 5: Lines If you have a function y(x), that has a constant derivative, m, and contains the point  then, using the accumulation idea I’ve been discussing in my last few posts, its equation is This is why I need your help! I want to ban all use of the slope-intercept form, y = mx + b, as a…