The applet shows the motion of a point P on the left, with a graph of its position f and velocity f' on the right. [br][list][*]Adjust the slider tool for n to create more data points along the graph of f. This simultaneously shrinks the width of the equal-sized sub-intervals of width [math]\Delta x[/math]. [/*][*]Use the checkboxes for f and f' to show the position and velocity graphs, respectively. [/*][*]Use the Slopes checkbox to show average velocity slopes on the graph of f. [/*][*]Use the AVG velocity checkbox to plot a new graph (a "step" function) whose value over an interval is the (constant) average velocity of the point over that interval. [/*][*]Use the Area checkbox to show the area "under" the curve over each interval. [/*][/list]

This lesson is about the definite integral [math]\int_a^bf'(x)dx[/math], which is essentially the net area "under" the graph of a derivative function f'(x). [br][br]If f'(x) represents a velocity, then [math]f'(x)\Delta x[/math] is essentially a very small change in position based on moving at a velocity of [math]f'(x)[/math] over a very small time [math]\Delta x[/math].[br][list][*]If velocity is positive, then these small changes in position are positive (i.e., position is increasing, moving upward). The area "under" the derivative is counted positively. [/*][*]If velocity is negative, then these small changes in position are negative (i.e., position is decreasing, moving downward). The area "under" (between the curve and the x-axis) is counted negatively. [/*][/list]If we add up all these little areas, each of which represents a small change in position, we get a total change in position over the time from x=a to x=b. So, the definite integral of a velocity function f'(x) over the interval [a, b] can be interpreted as a change in position (i.e., change in f(x)) from x = a to x = b. That is:[br][br][math]\int_a^bf'(x)dx=\Delta f=f(b)-f(a)[/math][br][br]Where we are going with this: This is called the Fundamental Theorem of Calculus (Part 2), and this gives us a way of evaluating definite integrals. But, to apply this theorem, we need to be able to start with a derivative f'(x) and find the function f(x) from which it is derived. We refer to this process as antidifferentiation and refer to f(x) as an antiderivative of f'(x). We will study this in more detail over the next two lessons.