Winter wk 2 – Thus.13.Jan.05 Review Calculus Ch.5.1-2: Distance traveled and the Definite integral Ch.5.3: Definite integral of a rate = total change Ch.5.4:

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Presentation transcript:

Winter wk 2 – Thus.13.Jan.05 Review Calculus Ch.5.1-2: Distance traveled and the Definite integral Ch.5.3: Definite integral of a rate = total change Ch.5.4: Theorems about definite integrals Energy Systems, EJZ

Review 5.1: Measuring distance traveled Speed = distance/time = rate of change of position v = dx/dt =  x/  t Plot speed vs time Estimate  x=v  t for each interval Area under v(t) curve = total displacement

Area under curve: Riemann sums Time interval = total time/number of steps  t = (b-a) / n Speed at a given time t i = v(t i ) Area of speed*time interval = distance = v(t)*  t Total distance traveled = sum over all intervals

Calc Ch.5-3 Conceptest

Calc Ch.5-3 Conceptest soln

Areas and Averages To precisely calculate total distance traveled x tot take infinitesimally small time intervals:  t  0, in an infinite number of tiny intervals: n   Practice: 5.3 #3, 4, 8 (Ex.5 p.240), 29

Ex: Problem 5.2 #20

Practice: Ch.5.4 #2

Analytic integration is easier Riemann sums = approximate: The more exact the calculation, the more tedious. Analytic = exact, quick, and elegant Trick: notice that

Analytic integration Total change in position Trick: Look at your integrand, v. Find a function of t you can differentiate to get v. That’s your solution, x! Ex: if v=t 2, then find an x for which dx/dt= t 2 Recall: so and x=

Practice analytic integration Total change in F = integral of rate of change of F 1. Look at your integrand, f. 2. Find a function of x you can differentiate to get f. 3.That’s your solution, F!

Symmetry simplifies some integrals Practice: Ch.5.4 # 16

Thm: Adding intervals

Calc Ch.5-4 Conceptest

Calc Ch.5-4 Conceptest soln

Thm: Adding & multiplying integrals Practice: Ch.5.4 # 4, 8

Thm: Max and min of integrals Practice: Ch.5.4 #