5.3: FTC Changed History
ID: 9778
Time required: 45 minutes
Activity Overview
This activity builds student comprehension of functions defined by a definite integral, where the independent variable is an upper limit of integration. Students are led to the brink of a discovery of a discovery of the Fundamental Theorem of Calculus, that
Topic: Fundamental Theorem of Calculus
 Graph a function and use Measurement > Integral to estimate the area under the curve in a given interval.
 Use Integral (in the Calculus menu) to obtain the exact value of a definite integral.
Teacher Preparation and Notes

This investigation should follow coverage of the definition of a definite integral, and the relationship between the integral of a function and the area of a region bounded by the graph of a function and the
x− axis. 
Before doing this activity, students should understand that if
a<b andf(x)>0 , then:
∫abf(x)>0 
∫ab−f(x)<0 
∫baf(x)<0 
∫ba−f(x)>0


Before starting this activity, students should go to the HOME screen and select
F6 :Clean Up > 2:NewProb, then press ENTER. This will clear any stored variables.
Associated Materials
 Student Worksheet: FTC Changed History http://www.ck12.org/flexr/chapter/9729, scroll down to the third activity.
Problem 1 – Constant integrand
Students explore the function
1. The table looks like the one below.



1  1.5 
2  3 
3  4.5 
4  6 
5  7.5 
2.
3. 1.5 units
4. The graph will be a line through the origin with slope 1.5.
Students will enter their data into the lists and then view the graph of
5. A line; yes (student answers may vary)
6. The same as before, except the slope would be 0.5 instead of 1.5.
Problem 2 – NonConstant Integrand
Students investigate the behavior of



1  0.25 
2  1 
3  2.25 
4  4 
5  6.25 
8.
9. The area changes by a different amount each time because both the height and width are increasing.
10. The graph is not linear. It is a parabola as seen by the formula in the above screen shot.
Problem 3 – An Integrand That Changes Sign



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

The fractions are approximated to the nearest hundredth.
11. After
12. The values for
13. The values for which the integral is increasing are
14. The table seems to indicate
15. Yes, we have seen a similar situation. The minimum occurs on a function where the function stops decreasing and starts increasing.
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