1. 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

2. 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

3. 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

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

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

6. 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

7. 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

8. 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

9. 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

10. 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

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