Calculus, ET First Edition Jon Rogawski Calculus, ET First Edition Chapter 6: Applications of the Integral Section 6.2: Setting Up Integrals: Volume, Density, Average Value Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

constant cross-sectional area. Its volume is then the product of its In this section, we will investigate using integrals to find the volume of objects of different shapes. In Figure 1, we see a right cylinder of constant cross-sectional area. Its volume is then the product of its cross-sectional area and its height: V = Ah. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

In Figure 2, we see a right cylinder whose cross-sectional area varies as some function of its height. If we were to imagine taking numerous horizontal slices through the cylinder, each slice would be a thin right cylinder whose volume could be approximated by: V = AΔh. In the limit, as Δh → 0, the sum of the individual volumes is the integral: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

More formally, Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Use an integral to find the volume of the pyramid in Figure 3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 389 2. Let V be the volume of a right circular cone of height 10 whose base is a circle of radius 4 (figure 16). (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

Find the volume of the solid in Figure 4, if the cross-sections area is a semi-circle. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Use the process from the previous slide to find an integral for the volume of sphere in Figure 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 389 8. Let B be the solid whose base is the unit circle x2 + y2 = 1 and whose vertical cross sections perpendicular to the x-axis are equilateral triangles . Show that the vertical cross sections have area and compute the volume of B. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 389 Find the volume of the solid with the given base and cross sections. 12. The base is a square, one of whose sides is the interval [0, l] along the x–axis. The cross sections perpendicular to the x–axis are rectangles of height f (x) = x2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework Homework Assignment #12 Review Section 6.2 Page 389, Exercises: 1 – 21(EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company