Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Jon Rogawski

Calculus, ETFirst Edition

Chapter 6: Applications of the IntegralSection 6.2: Setting Up Integrals: Volume,

Density, Average Value

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

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 itscross-sectional area and its height: V = Ah.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

In Figure 2, we see a right cylinder whose cross-sectional area variesas some function of its height. If we were to imagine taking numeroushorizontal slices through the cylinder, each slice would be a thinright cylinder whose volume could be approximated by: V = AΔh.

In the limit, as Δh → 0, the sum of the individual volumes is the integral:

b

aV f y dy

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

More formally,

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Use an integral to find the volume of the pyramid inFigure 3.

Example, Page 3892. 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 CalculusCopyright © 2008 W. H. Freeman and Company

Rogawski CalculusCopyright © 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 CalculusCopyright © 2008 W. H. Freeman and Company

Use the process from the previous slide to find an integral for the volume ofsphere in Figure 5.

Example, Page 3898. 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 CalculusCopyright © 2008 W. H. Freeman and Company

23 1A x x

Example, Page 389Find 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 CalculusCopyright © 2008 W. H. Freeman and Company

Homework

Homework Assignment #12 Review Section 6.2 Page 389, Exercises: 1 – 21(EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company