1. Homework Homework Assignment #12 Review Section 6.2
Page 389, Exercises: 1 – 33(EOO), 35
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 389
1. Let V be the volume of a pyramid of height 20 whose base is a square of side 8. (a) Use similar triangles to find the area of a horizontal cross section at a height y. (b) Calculate V by integration.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 389
5. Find the volume of liquid needed to fill a sphere of radius R to a height h (Figure 18).
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 389
Find the volume of the solid with the given base and cross sections. 9. The base is the unit circle x2 + y2 = 1 and the cross sections perpendicular to the x-axis are triangles whose heights and bases are equal.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 389
Find the volume of the solid with the given base and cross sections. 13. The base is the region enclosed by y = x2 and y = 3 and the cross sections perpendicular to the y-axis are squares.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 389
17. Find the volume V of a regular tetrahedron whose face is an equilateral triangle of side s.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 389
17. Find the volume V of a regular tetrahedron whose face is an equilateral triangle of side s.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 389
21. Figure 24 shows the solid S obtain by intersecting two cylinders of radius r whose axes are perpendicular. a) The horizontal cross section of each cylinder at distance y from the central axis is a rectangular strip. Find the strip’s width. b) Find the area of the horizontal cross section of S at distance y. c) Find the volume of S as a function of r.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

9. length, ρ, the mass will be: M = ρl, where l is the length.
If we wish to find the total mass of a rod of constant density per unit
length, ρ, the mass will be: M = ρl, where l is the length.
To find the total mass of the rod in Figure 6, we must use the integral:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

10. Example, Page 389
24. Find the total mass of a 1-m rod whose linear density function is ρ (x) = 10(x + 1)–2.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

11. For some urban planning functions, it might be appropriate to consider
the population density P as a function of distance r from the urban
center P (r).
To calculate the population living between r1 and r2 miles of the
urban area, we would use the integral:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

12. Example, Page 389
31. The density of deer in a forest is the radial function ρ (r) = 150(r2 + 2)–2 deer per km2, where r is the distance from a small meadow in km. Calculate the number of deer in the region 2 ≤ r ≤ 5 km.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

13. When calculating flow rates through piping systems, hydraulic
engineers usually use the expedient of “slug” flow, that is, they
assume that all of the fluid in a given section of the pipe travels
at the same rate v and the volume flowing through the pipe becomes:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

14. To calculate the flow rate used for slug flow calculations, engineers
use the laminar flow model which more closely approximates the
actual behavior of the fluid in the piping system. In laminar flow,
velocity of the fluid has some direct relationship to the distance
from the inner surface of the pipe, as shown in Figures 9 and 10.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

15. To calculate the flow rate Q using the laminar model where r is the
radius of the pipe measured from the center out, R is the maximum
radius, and v (r) is the flow rate as a function of distance from the
center of the pipe , we use the integral:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

16. Example, Page 389
35. A solid rod of radius 1 is placed in a pipe of radius 3so that their axis are aligned. Water flows through the pipe and around the rod. Find the flow rate if the velocity of the water 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

17. Homework Homework Assignment #12 Review Section 6.2
Page 389, Exercises: 1 – 33(EOO), 35
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company