Overview
In this section, we will take a look at the geometrical meaning of integrals with variables in their limits of integration, and see how this too is a type of function that we can analyse. These type of functions lead to the Second Fundamental Theorem of Calculus, which shows how we can define an antiderivative for any function.
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.

Evaluate integral functions such as \(A(x) = \int_{0}^{x}g(t) \; dt\) at various \(x\) values when given a simple formula or graph for \(g(t)\).

Evaluate integrals that have a variable in one of their limits.

State the Second Fundamental Theorem of Calculus and use it to state an antiderivative for a given function.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Draw the graph of \(f\), when given \(f'\) and an initial condition.

Interpret the physical and graphical meaning of a function in the form \(A(x) = \displaystyle\int_{0}^{x}f(t) \; dt\).

Use the Second FTC to differentiate an integral that has a variable in its limits.

Analyse an integral function using the usual techniques from differential calculus (those encountered in curve sketching).

Recognize the difference between a definite integral that evaluates to a number (such as \(\int_{1}^{2} x\; dx\)) and a definite integral that evaluates to a function (such as \(\int_{1}^{x} t \; dt\)).
To prepare for class

Watch the following video which shows how to sketch the graph of an antiderivative based on the graph of a function.

Do the Preview Activity for section 5.1 (on WeBWorK if required by your teacher).

Read all of section 5.1: Constructing accurate graphs of Antiderivatives_.

Read subsection 5.2 The second fundamental theorem of calculus and take good note of the statement of the theorem. We will go over the proof of the theorem again in class.

Do the Preview Activity for section 5.2 (on WeBWorK if required by your teacher).

Watch the following video which shows how to sketch the graph of a function defined as an integral:
After class

Watch the following videos:

Do some experimentation with the following interactive applets:
 Identify an Antiderivative Function (Marc Renault)

The Area Function (Marc Renault)

The Fundamental Theorem of Calculus, Part I (Theoretical Part) (Marc Renault)

The Fundamental Theorem of Calculus, Part II (Practical Part) (Marc Renault)

2nd Fundamental Theorem of Calculus (John Holden)