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www.le.ac.uk Integration – Volumes of revolution Department of Mathematics University of Leicester

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Content Around y-axisAround x-axisIntroduction

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If a curve is rotated around either the x-axis or y-axis, a solid is formed. The volume of this solid is called the “Volume of revolution”. Around y-axisAround x-axisNextIntroduction

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Examples : click to see the solids formed Around y-axisAround x-axisIntroductionNext

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y x Volume of Revolution around x-axis Around y-axisAround x-axisIntroductionNext

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x y Around y-axisAround x-axisIntroduction

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x Around y-axisAround x-axis y Introduction

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x y Around y-axisAround x-axisIntroductionNext

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Around y-axisAround x-axis Another way of looking at integration Introduction Click here to see what each bit means

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Around y-axisAround x-axis Another way of looking at integration Introduction

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Around y-axisAround x-axis Another way of looking at integration IntroductionNext ∫ means sum over all the strips

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x For a volume of revolution, we have circular chunks instead of strips. Around y-axisAround x-axisIntroductionNext

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Volume of Revolution around x-axis NextAround y-axisAround x-axisIntroduction

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Volume of Revolution around x-axis Example Let: Then on the interval 0 and 1: Around y-axisAround x-axisIntroductionNext

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x Volume of Revolution around y-axis Around y-axisAround x-axisIntroductionNext

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x y Around y-axisAround x-axisIntroduction

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x y Around y-axisAround x-axisIntroduction

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x y Around y-axisAround x-axisIntroductionNext

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Around y-axisAround x-axisIntroductionNext Volume of revolution around y-axis

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Around y-axisAround x-axisIntroductionNext

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Volume of revolution around y-axis Example Around y-axisAround x-axisIntroductionNext

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Find the following volumes of revolution: Around y-axisAround x-axisIntroductionNext

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Conclusion You should now be able to: Visualise the effect of rotating a shape around the x and y axes. Compute the volume of revolution. Further reading: try looking up the equations needed rotate a shape around the x-axis, this will require knowledge of polar coordinates. Around y-axisAround x-axisNextIntroduction

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6.3 – Volumes of Cylindrical Shells. Derivation Assume you have a functions similar to the one shown below and assume the f is to difficult to solve for.

6.3 – Volumes of Cylindrical Shells. Derivation Assume you have a functions similar to the one shown below and assume the f is to difficult to solve for.

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