Section 7.1: Integral as Net Change

Objective: Students will be able to…•Use the definite integral in a variety of applications

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?

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

2

1

)(t

t

dttv

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.

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 = 2

1

)(t

t

dttv

Linear Motion Example Still Continued

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

Wrap-up

• Integrating velocity = displacement

• Integrating absolute value of velocity = total distance traveled

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.

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?