Overview
Now that we're done studying techniques of integration, we can focus on applications: Integrals can be used to solve a wide variety of problems in physics, engineering, statistics, and other fields. This first section is an introduction to the ideas behind these applications, showing how integrals can calculate more than just the net area between a curve and the \(x\)axis.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.

(Algebra review) Given two functions, graph them both, calculate their points of intersection, and identify intervals in which one function is always above (or below) the other.

Given two functions \(f\) and \(g\) such that \(g(x) \geq f(x)\) on \([a,b]\), set up and evaluate a definite integral that represents the area between them.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Given two functions, possibly functions of \(y\), calculate the area between them by choosing the proper differential (\(dx\) or \(dy\)) and setting up a definite integral with the correct variable.

Given two functions, break the area between them into two or more sections based on their intersection points, set up, and evaluate an integral representing each area.

Given a differentiable function \(f\), set up and evaluate a definite integral which gives the length of the curve \(y = f(x)\) from \(x = a\) to \(x = b\).
To Prepare for Class

Do the first problem of the Preview Activity for this section (on WeBWorK if required by your teacher).

Watch the following short video which gives you a short overview of all the important concepts in this section:

Watch the following videos which show and explain in more detail how to compute the area between two curves by slicing the area into vertical rectangles, and thus integrating with respect to the \(x\) variable.

Do the second problem of the Preview Activity for this section (on WeBWorK if required by your teacher).

Watch the following video which shows and explains how to compute the area between two curves by slicing the area into horizontal rectangles, and thus integrating with respect to the \(y\) variable.

Watch the following video which shows an example of finding the length of a curve: