MATH1014_Volume_cylindrical_shells.pdf - Applications of Integration Volumes by cylindrical sheels(Mainly based on Stewart Chapter 6 \u00a76.3 Edmund Chiang

MATH1014_Volume_cylindrical_shells.pdf - Applications of...

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Applications of Integration: Volumes by cylindrical sheels (Mainly based on Stewart: Chapter 6, § 6.3) Edmund Chiang MATH1014 September 9, 2019 1 Volumes 1.1 Volumes by cylindrical shells We consider volumes that are difficult or even impossible to find with the techniques from the previous sections. For example, let us consider the following problem. Example . Find the volume of the solid obtained by rotating about the region y = 2 x 2 - x 3 and y = 0 about the y - axis. Figure 1: (Stewart: Volume of y = 2 x 2 - x 3 by revolution about the y - axis) If we apply the method used in the previous section to find area of vertical thin strip, then given y , we need to know the corresponding inner and outer radii of the solid. That is, we need to solve the cubic equation y = 2 x 2 - x 3 for x . But the cubic formula is very complicated to be really useful. Moreover, if the relation between y and x is of fifth degree or higher, then the celebrated Galois theory from the 19th century already asserted that no such formula for x can be found in general. Thus we look for alternative method instead.

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