Definite Integral and Area: A Comparison   

 *** VIDEO with explanation ***  Class Notes:  Definite Integral & FTC       Homework         Home                    

The definite integral over an interval of a non-negative portion of a function can be interpreted as area.

For the function below, a is the lower limit and b is the upper limit of the interval. Click and drag the sliders to change the values for a and b. Notice the change in the area beneath the curve.

Since area is never negative, we can only say that this definite integral represents area for the portions of the function that are not negative.
However, we can consider definite integrals for intervals that include negative portions of a function. But it is not interpreted as area.

Click and drag the b slider to increase the value of b to b=5.
What happens to the value of the definite integral, I, as b changes from 4 to 5?  ________________________
 Notice that this definite integral no longer represents area.

Using the sliders, find the area under the curve, y = -x² + 4x, from a=0 to b=4. Then find the definite integral from a=4 to b=5.
How does the sum of these two values
compare to the integral of the the function from a=0 to a=5?    ______________________

With a = 0, set b=0.  Then right click on the b slider and set to animation on.
Watch the value of I, the definite integral, as it changes automatically, as b increases to b=4 and then beyond 4 to b=5.
When does I begin to decrease?    ____________      Why?   __________

G. Battaly, Math Department, Westchester Community College, 29 April 2013, Created with GeoGebra     
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