We present part of a project whose aim it is to investigate high school students’ meanings for fundamental concepts of calculus, including rate of change and accumulation. We focus on students’ reactions to a peer’s statement that an integral or accumulated quantity can be found by adding up vertical lines under the graph of the rate of change function being integrated. We present results of 24 interviews with advanced level high school students after they studied integration. Our findings show that most of the interviewees consider the peer’s thinking as correct in principle. On the other hand, more than half of the interviewees state that the peer’s way of thinking is not close to their own. A tentative conclusion is that most of these advanced-level high school students proficiently carry out computations of integrals but fail to relate to the essence of a definite integral as a sum of products.