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!)
Good Question 6 – 2000 AB 4 One of my favorite questions
Real “Real Life” Graph Reading
Jobs, Jobs, Jobs are real life accumulation problem based on a graph
Graphing with Accumulation 1 Increasing and decreasing.
Graphing with Accumulation 2 Concavity
Accumulation and Differential Equations Differential equations will be considered next week, but this idea relates to integration.
Rate and Accumulation Questions (Type 1)
Teaching Calculus 