Section 7.1: Integral as Net Change

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Looking at position, velocity, and acceleration from the integral.

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Presentation transcript:

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

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 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 =

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?