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Math 181

6.1 – Volumes Using Cross-Sections

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𝑛→∞∑𝑘=1

𝑛

𝐴 (𝑐𝑘 ) Δ𝑥 ∫0

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𝐴(𝑥)𝑑𝑥

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The volume of a solid with cross-sectional areas from to is:

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Ex 1.A solid lies between planes perpendicular to the -axis at and . The cross-sections perpendicular to the -axis are circular disks whose diameters run from the parabola to the parabola . Find the volume of the solid.

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Ex 2.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis., , ,

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Ex 2.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis., , ,

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Ex 2.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis., , ,

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Ex 2.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis., , ,

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Ex 3.Find the volume of the solid generated by revolving the region between the -axis and the curve , , about the -axis.

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Ex 3.Find the volume of the solid generated by revolving the region between the -axis and the curve , , about the -axis.

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Ex 3.Find the volume of the solid generated by revolving the region between the -axis and the curve , , about the -axis.

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Ex 3.Find the volume of the solid generated by revolving the region between the -axis and the curve , , about the -axis.

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Ex 4.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis.,

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Ex 4.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis.,

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Ex 4.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis.,

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Ex 4.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis.,

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Ex 4.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the -axis.,

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Ex 5.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the line .,

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Ex 5.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the line .,

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Ex 6.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the line .,

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Ex 6.Find the volume of the solid generated by revolving the region bounded by the following lines and curves about the line .,

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Practice Problems

- Work individually…

- …but feel free to get help from neighbors.

- Raise hand to get help from me.

- Remember: the harder you work here, the easier your homework will be.

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Practice Problems

- Work individually…

- …but feel free to get help from neighbors.

- Raise hand to get help from me.

- Remember: the harder you work here, the easier your homework will be.

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Practice Problems

- Work individually…

- …but feel free to get help from neighbors.

- Raise hand to get help from me.

- Remember: the harder you work here, the easier your homework will be.

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Practice Problems

- Work individually…

- …but feel free to get help from neighbors.

- Raise hand to get help from me.

- Remember: the harder you work here, the easier your homework will be.

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Practice Problems

- Work individually…

- …but feel free to get help from neighbors.

- Raise hand to get help from me.

- Remember: the harder you work here, the easier your homework will be.