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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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