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Volumes

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Right Cylinder

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Volume of a Right Cylinder (Slices) Cross section is a right circular cylinder with volume Also obtained as a solid of revolution Consider the case when h=5 and r=2

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Volume of a Rectangular Box Cross section is a right cylinder with volume Consider the case of a box with dimensions 3x5x7 Area of a cross section is b*l

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Volume of a Sphere of radius r (slices) Cross section is a right cylinder with volume Consider the case of a sphere of radius r=3

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Another Example Sketch the solid of revolution obtained by rotating around x-axis and find its volume Sketch the solid of revolution obtained by rotating around y-axis and find its volume Sketch the solid of revolution obtained by rotating around x=5 and find its volume

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Volume using Shells

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Example Sketch the solid of revolution obtained by rotating about x-axis and find its volume Sketch the solid of revolution obtained by rotating around y-axis and find its volume Sketch the solid of revolution obtained by rotating around x=1.2 and find its volume

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Cylindrical Shells Definition of a shell Animations

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Formulas Shell in general position Volume of shell

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Volume Formulas Function y=f(x) Rotating about y-axis Function x=f(y) Rotating about x-axis

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- Volumes of a Solid The volumes of solid that can be cut into thin slices, where the volumes can be interpreted as a definite integral.

- Volumes of a Solid The volumes of solid that can be cut into thin slices, where the volumes can be interpreted as a definite integral.

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