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MAT 1235 Calculus II Section 5.3 Volumes by Cylindrical Shells

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Presentation on theme: "MAT 1235 Calculus II Section 5.3 Volumes by Cylindrical Shells"— Presentation transcript:

1 MAT 1235 Calculus II Section 5.3 Volumes by Cylindrical Shells http://myhome.spu.edu/lauw

2 Homework WebAssign HW 5.3 It is crucial to have a correct (mental) images.

3 Tomorrow Section 6.1 Inverse Functions Review your algebra/pre-calculus… What is an inverse function? What is a one-to-one function? Also, we will talk about possible scheduling changes next week.

4 Preview 5.2 Find volumes by the “Disc” method 5.3 Find volumes by the “Shell” method Can handle cases where the Disc method does not work well.

5 Recall Any limit of the form : can be interpreted as

6 Example 1 (Disc Method)

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10 Cylindrical Shells

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12 use midpoint as the sample point

13 Cylindrical Shells use midpoint as the sample point

14 Cylindrical Shells

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17 Shell Height Radius of the Shell

18 Example 1 (Shell Method) Find the volume of the solid obtained by rotating about the y-axis the region bounded by and

19 Example 1

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21 Example 2 Find the volume of the solid obtained by rotating about the y-axis the region bounded by and

22 Example 2

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25 Remark:

26 Example 3

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29 Remark:

30 Example 4

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