1. Section 7.1: Integral as Net Change
Objective: Students will be able to…
Use the definite integral in a variety of applications

2. Linear Motion Revisted
A particle moves along the x-axis so that its velocity at time t is v(t) = t2 – t – 6 m/sec, 1 ≤ t ≤ 4
a) Describe the motion. When does it move to the right, to the left, and when is it stopped?

3. Displacement The change in an object’s position
Displacement = rate of change x time
Displacement =
Displacement is negative: moving left
Displacement is positive: moving right

4. Linear Motion Example Continued
For v(t) = t2 – t – 6 m/sec, 1 ≤ t ≤ 4, find the particle’s displacement for the given time interval.

5. Total Distance Total Distance =
When finding total distance traveled, consider the intervals where the particle moves to the left and the intervals where the particle moves to the right.
Total Distance =

6. Linear Motion Example Still Continued
Find the total distance traveled by the particle. v(t) = t2 – t – 6 m/sec, 1 ≤ t ≤ 4

7. Wrap-up Integrating velocity = displacement
Integrating absolute value of velocity = total distance traveled

8. More Applications
The integral of the rate of change of any quantity gives the net change in that quantity.
Example:
Let F(t) represent a bacteria population which is 5 million at time t = 0. After t hours, the population is growing at an instantaneous rate of 2t million bacteria per hour. Estimate the total increase in the bacteria population during the first hour and the population at t = 1.

9. Ready for another?
A pizza heated to a temperature of 350⁰ F is taken out of an oven and placed in a 75⁰ F room at time t = 0 minutes. The temperature of the pizza is changing at a rate of -110e-0.4t degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the pizza at time t = 5 minutes?