1. Homework Homework Assignment #13 Review Section 6.2 Page 389, Exercises: 25 – 33(EOO), 35 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 389 25.Find the mass of a 2-m rod whose linear density function is ρ ( x ) = 1 + 0.5 sin ( πx ) kg/m for 0 ≤ x ≤ 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 389 29. Table 1 lists the population density (in people per square km) as a function of distance r (in km) from the center of a rural town. Estimate the total population within a 2-km radius of the center by taking the average of the left- and right-endpoint approximations. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 389 33.Find the flow rate through a tube of radius 4 cm, assuming that the velocity of fluid particles at a distance r cm from the center is v (r) = 16 – r 2 cm/s. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 389 35.A solid rod of radius 1 cm is placed in a pipe of radius 3 cm so their axes are aligned. Water flows through the pipe and around the rod. Find the flow rate if the velocity is given by the radial function v (r) = 0.5(r – 1)(3 – r) cm/s. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 389 35.Continued. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Average Values If we drive for five hours and cover 300 miles, we would say our average speed was 60 mph. Graphically, it might look like this: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Remembering that the integral gives us the area between the graph of the function and the x-axis on the interval [a, b], dividing the area by the width (b – a) will give us the average value of f (x) on [a, b

9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 12, the area under the graph of f (x) = sin x on [0, π] is the same as the are of the rectangle with length π and width 2/π.

10. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Example, Page 389 Calculate the average over the given interval. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Example, Page 389 Calculate the average over the given interval. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Example, Page 389 58.Let M be the average value of f (x) = x 4 on [0, 3]. Find a value of c in [0, 3] such that f (c) = M. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Homework Homework Assignment #14 Read Section 6.3 Page 389, Exercises: 37 – 59(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company