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

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Integration – Volumes of revolution Department of Mathematics University of Leicester

Content Around y-axisAround x-axisIntroduction

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

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

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

x y Around y-axisAround x-axisIntroduction

x Around y-axisAround x-axis y Introduction

x y Around y-axisAround x-axisIntroductionNext

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

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

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

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

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

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

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

x y Around y-axisAround x-axisIntroduction

x y Around y-axisAround x-axisIntroduction

x y Around y-axisAround x-axisIntroductionNext

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

Around y-axisAround x-axisIntroductionNext

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

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

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