1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Estimating with Finite Sums Section 5.1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 2 What you’ll learn about Distance Traveled Rectangular Approximation Method (RAM) Volume of a Sphere Cardiac Output … and why Learning about estimating with finite sums sets the foundation for understanding integral calculus.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Distance Traveled A train moves along a track at a steady rate of 75 miles per hour from 7:00 AM to 9:00 AM. What is the total distance traveled by the train? Slide 5- 3

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall If the velocity remains the same, a rectangle area can be made no matter how fast the train is going or how long or short the time interval was. What if the train had a velocity that varied as a function of time? Would the area of the irregular region still give the total distance traveled over the time interval? Slide 5- 4

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Mathematicians thought that it would, so they were interested in a calculus for finding areas under curves. They imagined the time interval being partitioned into many tiny subintervals (sliced into narrow strips), nearly indistinguishable. Slide 5- 5

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 6 Example Finding Distance Traveled when Velocity Varies A particle starts at x = 0 and moves along the x-axis with velocity v(t) = t 2 for time t > 0. Where is the particle at t = 3?

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Rectangular Approximation Method (RAM) Midpoint Rectangular Approximation Method (MRAM) Left-hand Rectangular Approximation Method (LRAM) Right-hand Rectangular Approximation Method (RRAM) No matter what the method, add products of the form f(x i )  x Slide 5- 7

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 8 LRAM, MRAM, and RRAM approximations to the area under the graph of y=x 2 from x=0 to x=3 Animated

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall If the integrand is increasing, then all LRAMs underestimate the exact value of the integral and all RRAMs overestimate the integral. If the integrand is decreasing, then all LRAMs overestimate the exact value of the integral and all RRAMs underestimate the integral. Slide 5- 9

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 10 Example Finding Distance Traveled when Velocity Varies A particle starts at x = 0 and moves along the x-axis with velocity v(t) = t 2 for time t > 0. Where is the particle at t = 3?

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 11 Example Finding Distance Traveled when Velocity Varies

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 12 Example Estimating Area Under the Graph of a Nonnegative Function Estimate the area under the graph of f(x) = x 2 sin x from x = 0 to x = 3. See it

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 13 Example Estimating the Volume of a Sphere Estimate the volume of a solid sphere of radius 4.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 14 Example Computing Cardiac Output from Dye Concentration Estimate the cardiac output of the patient whose data appears on page 252 and Figure 5.11. Give the estimate in liters per minute.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Pages 254-255 (5, 9, 11) Slide 5- 15