5.5 Net Change as the Integral of a Rate Mon Nov 30 Do Now Find the area under each function 1) f(x) = sin x over [0,pi] 2) g(x) = x^2 – x + 1 over [0, 2]

Quiz Review Make up quizzes by Friday

Net change as an integral Area under a curve is not the only thing that can be measured by a definite integral Any accumulation can be expressed as a definite integral – Distance traveled over time – Volume built up or leaked over time – Basically any measurement over time

Net change The net change in f(t) over an interval [t1,t2] is given by the integral If we have a rate, we can use an integral to measure how much was accumulated over a time interval

Ex 1 Water leaks from a tank at a rate of 2 + 5t liters per hour, where t is the number of hours after 7 AM. How much water is lost between 9 and 11 AM?

Ex 2 In Book p.323

Integral of Velocity = Position When we talk about position and velocity, it is important to know the difference between displacement and distance traveled Displacement can be 0 if you return to starting point, while distance traveled would not be 0

Integral of Velocity For an object in linear motion with velocity v(t), then Displacement during [t1,t2] = Distance traveled during [t1,t2] =

Ex A particle has velocity m/s. Compute the displacement and total distance traveled over [0,6]

Closure Which of the following would be represented as derivatives and which as integrals? – Velocity of a train – Rainfall during a 6 month period – Mileage per gallon of an automobile – Increase in US population from 1990 to 2010 HW: p.326 #

5.5 Net Change as the Integral of a Rate Tues Feb 10 Do Now Find the displacement and distance traveled of a particle that moves in a straight line with the velocity v(t) = t + 3t^2 on the interval [0, 10]