5.4 Indefinite Integrals and the Net Change Theorem

1-4. Verify by differentiation that the formula is correct.

1. .

 Let . Then .

      That is

2. .

 Let and . Then and by differentiation.

      That is

3. .

 Let then .

      So

      Also change of variable for , then .

      That is

4. .

 Let then .

      So

                             .

      Also change of variable for , then .   That is

5-13. Find the general indefinite integral.

5. .

 .

6. .

 .

7. .    

 .

8. .

 .

9. .

 .

10. .     

11. .

 .

12. .

 .

13.      .

        .

      Let . Then .

      So, .

14-22. Evaluate the integral.

14. . 

.

15. .    

16. .   

17. . 

18. .

19. .  

20. .

http://matrix.skku.ac.kr/cal-lab/cal-5-4-20.html

-1/2*sqrt(2)*log(-2*sqrt(6) + 5) + sqrt(2)*sqrt(6)

21. .

22. .

 Let . Then

      Thus, .

23. Estimate the area of the region that lies under the curve and above the -axis.

http://matrix.skku.ac.kr/cal-lab/cal-5-4-23.html

0.876956449003

24. Water is being added to a tank at a rate of per minute. How much water is added to the tank from to ?

 The amount of water added for is where

      so that .

25. Find the area of the region that lies to the right of the -axis and to the left of the parabola .

 Since the parabola meets the -axis at and , the area is given by

      .

26. A particle is moving along a line with the velocity function . Find the dis-placement and the distance traveled by the particle during the time .

 The displacement = , the distance .

27. A honeybee population starts with 30 bees and increases at a rate of bees per week. How many honeybees are there after 10 weeks?

 Since the net change in population during 10 weeks is , the total number of honeybees after 10 weeks is .

28. The acceleration function (in ) of a particle is given by and the initial velocity is .

    Find the velocity of the particle at time and determine the total distance traveled for .

 Since , the velocity at time is . Therefore the distance traveled for is .