MA180/186/190 Calculus Semester 2. Week 1: What is Integration about?
In Calculus, integration and differentiation are (in a way) opposite sides of the same coin, but that is not the only motivation to study integration. Another is because it gives us a means of calculating areas under curves defined by algebraic equations, which is something that we often want to know in applied contexts. The Irish word for integration is “suimeáil”, which means “summing”, and this word maybe captures the meaning more than the English word “integration” does. One thing that integration is good for is calculating “totals” produced by processes that are varying continuously. If you want to calculate the total distance that you ran this week and you know how far you ran each day, that is an addition problem and a relatively easy task. But if you want to calculate the total volume of water that entered a reservoir this week, and what you have is a continuously varying record of the instantaneous flow past a particular point in a pipe, that is not exactly an addition problem – you can’t add up the values over an infinite number of “instants”. What you need is the concept of a (definite) integral.
Our goal for the first week is to understand this concept of a definite integral as an extension of the (simpler) concept of a sum, and to become familiar with the special notation that is used to write about integrals.
Relevant sections of the Lecture notes this week are Sections 1.1 and 1.2.
Here is Lecture 1 (Wednesday January 12)
Here is Lecture 2 (Thursday January 13).
Weekly Problem 1
The weekly problems are just for fun. They have nothing much to do with our curriculum. Please send me an email if you have a solution that you would like to share with the class!