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 axis.

Before doing this activity, students should understand that if and , then:
 Before starting this activity, students should go to the HOME screen and select :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 . They should notice that there is a constant rate of change in the graph of This rate of change is 1.5.
1. The table looks like the one below.
1  1.5 
2  3 
3  4.5 
4  6 
5  7.5 
2. ; There is zero area under the graph of from to .
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 . Students should note that this function changes at a nonconstant rate and are asked to explain why this is so (from a geometric point of view).
1  0.25 
2  1 
3  2.25 
4  4 
5  6.25 
8. The height and the length of the triangle are 0 so the area is 0.
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 , the integral value begins to decrease.
12. The values for in which the integral decreases are ; the function is negative.
13. The values for which the integral is increasing are ; the function is positive.
14. The table seems to indicate . To find out for sure, use fMin on the integral. You have to restrict the domain to , to get the answer.
15. Yes, we have seen a similar situation. The minimum occurs on a function where the function stops decreasing and starts increasing.