Calculus 6.

6 Properties of Definite Integrals Notes

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𝑓 𝑥 1. 𝑓 𝑥 𝑑𝑥

2. 𝑓 𝑥 𝑑𝑥

Properties of Definite Integrals

Equivalent Limits Reversal of Limits
𝒃

𝑓 𝑥 𝑑𝑥 𝟎 𝑓 𝑥 𝑑𝑥 𝒇 𝒙 𝒅𝒙
𝒂

Multiply by constant Adjacent Intervals

(𝒌 constant) (𝒂 𝒄 𝒃)

𝒃 𝒃

𝑘𝑓 𝑥 𝑑𝑥 𝒌 𝒇 𝒙 𝒅𝒙 𝑓 𝑥 𝑑𝑥 𝑓 𝑥 𝑑𝑥 𝒇 𝒙 𝒅𝒙
𝒂 𝒂

Addition Subtraction
𝒃 𝒃 𝒃 𝒃

𝑓 𝑥 𝑔 𝑥 𝑑𝑥 𝒇 𝒙 𝒅𝒙 𝒈 𝒙 𝒅𝒙 𝑓 𝑥 𝑔 𝑥 𝑑𝑥 𝒇 𝒙 𝒅𝒙 𝒈 𝒙 𝒅𝒙
𝒂 𝒂 𝒂 𝒂

𝑓 𝑥 3. 𝑓 𝑥 𝑑𝑥

4. 3𝑓 𝑥 𝑑𝑥

5. Given that 𝑓 𝑥 𝑑𝑥 4, 𝑓 𝑥 𝑑𝑥 3, and 𝑔 𝑥 𝑑𝑥 8, find the following.

a. 𝑓 𝑥 𝑑𝑥 b. 𝑓 𝑥 𝑑𝑥 c. 𝑓 𝑥 2𝑔 𝑥 𝑑𝑥

d. 𝑓 𝑥 𝑑𝑥 e. 3𝑓 𝑥 𝑑𝑥 f. 𝑓 𝑥 𝑔 𝑥 𝑑𝑥
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Piecewise-functions and integrals
6. The function 𝑔 is defined by 7. What is the value of |𝑥 2| 𝑑𝑥?
3 for 𝑥 2
𝑔 𝑥
4 𝑥 for 𝑥 2
What is the value of 𝑔 𝑥 𝑑𝑥?

What is the difference between 𝑓 𝑥 𝑑𝑥 and |𝑓 𝑥 | 𝑑𝑥?

y


 8. 𝑓 𝑥 𝑑𝑥

      9. |𝑓 𝑥 | 𝑑𝑥



Using a calculator to find an integral value

Sketch a graph of the definite integral. Use the calculator to evaluate.
10. √𝑥 1 𝑑𝑥 11. 1 𝑑𝑥
6.6 Properties of Definite Integrals
Calculus
Practice
The graph of f consists of line segments and a semicircle. Evaluate each definite integral.
1.
a. 𝑓 𝑥 𝑑𝑥 d. 𝑓 𝑥 𝑑𝑥

𝑓 𝑥

b. 𝑓 𝑥 𝑑𝑥 e. 𝑓 𝑥 𝑑𝑥

c. 2𝑓 𝑥 𝑑𝑥 f. |𝑓 𝑥 | 𝑑𝑥

Sketch a graph of the definite integral. Evaluate the integral with a graphing calculator.

2. √𝑥 1 𝑑𝑥 3. |𝑥 1| 𝑑𝑥 4. 1 𝑑𝑥

Let 𝒇 and 𝒈 be continuous functions that produce the following definite integral values.

𝑓 𝑥 𝑑𝑥 2 𝑓 𝑥 𝑑𝑥 5 𝑔 𝑥 𝑑𝑥 6

Find the following.

5. 2𝑓 𝑥 𝑑𝑥 6. 4 𝑓 𝑥 𝑑𝑥 7. 𝑓 𝑥 𝑑𝑥 8. 𝑔 𝑥 𝑑𝑥

9. 𝑔 𝑥 𝑓 𝑥 𝑑𝑥 10. 𝑓 𝑥 𝑑𝑥 11. 𝑓 𝑥 𝑑𝑥
 Let 𝒇 and 𝒈 be continuous functions that produce the following definite integral values.

𝑓 𝑥 𝑑𝑥 2 𝑓 𝑥 𝑑𝑥 4 𝑔 𝑥 𝑑𝑥 8

Find the following.

12. 𝑔 𝑥 𝑑𝑥 13. 𝑔 𝑥 𝑑𝑥 14. 3 𝑓 𝑥 𝑑𝑥 15. 𝑓 𝑥 𝑑𝑥

16. 𝑓 𝑥 𝑔 𝑥 𝑑𝑥 17. 3𝑓 𝑥 𝑔 𝑥 𝑑𝑥 18. |𝑓 𝑥 𝑔 𝑥 | 𝑑𝑥 19. 𝑓 𝑥 𝑔 𝑥 𝑑𝑥

6.6 Properties of Definite Integrals Test Prep

 y

20. 

𝑓 𝑡
f(x)


𝑡x
           







The graph of the function 𝑓 is shown above. Let 𝑔 be the function defined by 𝑔 𝑥 𝑓 𝑡 𝑑𝑡.
a. Find the average rate of change of 𝑔 from 𝑥 4 to 𝑥 6.

b. Find the instantaneous rate of change of 𝑔 with respect to 𝑥 at 𝑥 5, or state that it does not exist.

c. On what open intervals, if any, is the graph of 𝑔 concave down? Justify your answer.

d. Find all 𝑥-values in the interval 4 𝑥 6 at which 𝑔 has a critical point. Classify each critical point as
the location of a local minimum, a local maximum, or neither. Justify your answers.