Calculus-Sec-5-4-Solution

5.4   Indefinite Integrals and the Net Change Theorem

by SGLee - HSKim- SWSun

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

1.

We take derivative of the right hand side and show that it is integrand.

2.

We take derivative of the right hand side and show that it is integrand.

3.

Let . Then  and       (because  and )

But

Therefore.

4.

Let . Then  and  .

Substitute , then . Therefore

.

5-13. Find the general indefinite integral.

5.

.

6.

.

7.

.

8.

.

9.

.

10.

.

11.

.

12.

.

13.

.

14-22. Evaluate the integral.

14.

.

15.

16.

17.

18.

19.

.

21.

22.

.

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

It is easy to see that the graph has a real root 1.1914.

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   and .

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 .

And since .

Therefore the distance traveled for  is .