Integration (The Concept of Accumulation)

Most students know that if you are walking at a constant velocity, then the distance that you walk is equal to your velocity times the amount of time that you walk, that is, s = v t. Thus, if v = 4 ft/sec and t = 5 seconds, then s = 4(5) = 20 ft.

Plot v(t) on 0t5 and -5v30.

Note that the area of the rectangular region under the velocity line is 4(5) = 20. Thus, you can associate the total distance walked with the area under the plot of the velocity function.

How far do you walk after 1 second? 2 seconds? 3 seconds? 4 seconds? t seconds?

To visualize the distance that you walk as a function of time,

Click the Integration tool on the Tool Palette in the Graph window.
Click in the upper rectangular region of the Integration tool's icon to view the pop-up menu.
Select "Dynamic" from the pop-up menu.
Click the Go button.

Watch the area (total distance walked) under the plotted velocity function accumulate as a function of time. Also, note that the value the total "area so far" under the curve is being plotted as the area is being accumulated. Thus, the plotted line represents the distance that you walked as a function of time.

Note: You can click the Step button to move one time increment at a time, you can click the Stop button to stop the Dynamic Integrator, or you can click the Erase button to erase the drawn area and start a new plot. Also, you can increase or decrease the speed of the Dynamic Integrator by selecting "Dynamic Plotting Speed..." from the Options menu.

What is the equation of the line plotted by the Dynamic Integrator?

Let's extend the above concept of "distance walked as the total accumulated area under the velocity curve" to velocity functions that are not constant. As an example, suppose that your velocity increases linearly as you walk, that is, v = t + 2.

Plot vl(t) on 0t5 and -5v30.
Click the Go button.

Can you guess the type of function represented by the plotted distance curve?

What is the trapezoidal area under the velocity line between 0 and t? Hint: use the formula for finding the area of a trapezoid.

Did you guess that s(t) = t^2/2 + 2t?

Negative Velocities

What happens if at some point in time your velocity is negative, that is, at some point in time you are walking backwards. To find out, let's analyze the quadratic velocity function vq(t) = 2(t–1)(t–3.5) .

Plot vq(t) on 0t5 and -5v30.
Click the Go button.

Can you guess the type of function represented by the plotted "distance" curve?

Does the plotted "distance" curve represent the total distance walked? If not, what does the curve plotted by the Dynamic Integrator represent? What does the area "under" the velocity curve where the curve is below the t-axis represent? How could you find the total distance walked?

Piecewise Velocity Function

Plot vp(t) on 0t5 and -5v30.
Click the Go button.

Where is the distance function increasing at a constant rate?

Where is the distance function increasing the fastest?

Where is the distance function increasing the slowest?

Guessing the Form of the Distance Function (Antiderivatives)

Plot vc(t) on 0t10 and -5v30. Note the change in the t plotting domain.
Click the Go button.

Note that the curve plotted by the Dynamic Integrator does not represent the total distance you walked in t secs, but, it represents your distance at time t from your starting point.

Can you guess the function that is represented by the curve plotted by the Dynamic Integrator?

Select New Function from the Work menu.
Enter the expression for your guessed function into the first cell of the Work window. Press the Enter key.
Select Overlay Plot from the Graph menu. Does your function match the curve plotted by the Dynamic Integrator? If not, edit your function expression and try again.

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