## Areas, Integrals, and Antiderivatives

Read this section to learn about the relationship among areas, integrals, and antiderivatives. Work through practice problems 1-5.

### Integrals, Antiderivatives, and Applications

The antiderivative method of evaluating definite integrals can also be used when we need to find an "area", and it is useful for solving applied problems.

**Example 5: **A robot has been programmed so that when it starts to move, its
velocity after seconds will be feet/second.

(a) How far will the robot travel during its first 4 seconds of movement?

(b) How far will the robot travel during its next 4 seconds of movement?

(c) How many seconds before the robot is 729 feet from its starting place?

Solution:

(a) The distance during the first 4 seconds will be the area under the graph (Fig. 8) of velocity , from to , and that area is the definite integral . An antiderivative of feet.

(c) This part is different from the other two parts. Here we are told the lower integration endpoint, , and the total distance, 729 feet, and we are asked to find the upper endpoint. Calling the upper endpoint , we know that seconds.

**Practice 4: **(a) How far will the robot move between second and seconds?
(b) How many seconds before the robot is 343 feet from its starting place?

**Example 6: **Suppose that minutes after putting 1000 bacteria on a Petri plate the rate of growth of the population is 6t bacteria per minute. (a) How many new bacteria are added to the population during the first 7 minutes? (b) What is the total
population after 7 minutes? (c) When will the total population be 2200 bacteria?

Solution:

(a) The number of new bacteria is the area under the rate of growth graph (Fig. 9), and one antiderivative of is (check that ) so new bacteria = .

(b) The new population = {old population} + {new bacteria} = 1000 + 147 = 1147 bacteria

(c) If the total population is 2200 bacteria, then there are 2200 – 1000 = 1200 new bacteria, and we need to find the time T needed for that many new bacteria to occur.

1200 new bacteria = and minutes. After 20 minutes, the total bacteria population will be 1000 + 1200 = 2200.

**Practice 5: **(a) How many new bacteria will be added to the population between and minutes? (b) When will the total population be 2875 bacteria? (Hint: How many are new?)