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Understanding Distance Traveled and Definite Integrals in Calculus

This review covers key concepts from Calculus Chapter 5, focusing on distance traveled, the definition of a definite integral, and theorems related to integrals. It explores how to calculate total distance using speed as a function of time, introduces Riemann sums, and emphasizes the importance of analytic integration. Through practice problems and concepts, learners will gain a comprehensive understanding of measuring changes in position, calculating total distance over time intervals, and utilizing symmetry in integration for simplification.

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Understanding Distance Traveled and Definite Integrals in Calculus

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  1. 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

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

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

  4. Calc Ch.5-3 Conceptest

  5. Calc Ch.5-3 Conceptest soln

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

  7. Ex: Problem 5.2 #20

  8. Ex: Problem 5.2 #20 Practice: Ch.5.4 #2

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

  10. 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=t2, then find an x for which dx/dt= t2 Recall: so and x=

  11. 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. • That’s your solution, F!

  12. Symmetry simplifies some integrals Practice: Ch.5.4 # 16

  13. Thm: Adding intervals

  14. Calc Ch.5-4 Conceptest

  15. Calc Ch.5-4 Conceptest soln

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

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

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