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

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1 Integration – Volumes of revolution Department of Mathematics University of Leicester

2 Content Around y-axisAround x-axisIntroduction

3 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

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

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

6 x y Around y-axisAround x-axisIntroduction

7 x Around y-axisAround x-axis y Introduction

8 x y Around y-axisAround x-axisIntroductionNext

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

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

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

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

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

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

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

16 x y Around y-axisAround x-axisIntroduction

17 x y Around y-axisAround x-axisIntroduction

18 x y Around y-axisAround x-axisIntroductionNext

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

20 Around y-axisAround x-axisIntroductionNext

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

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

23 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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