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

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Rogawski Calculus Copyright © 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 its cross-sectional area and its height: V = Ah.

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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:

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

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

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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

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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.

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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.

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Example, Page 389 8.Let B be the solid whose base is the unit circle x 2 + y 2 = 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

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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) = x 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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

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