1. © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 29: Volumes of Revolution

2. Volumes of Revolution "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C3 AQA Edexcel Module C4 OCR MEI/OCR

3. Volumes of Revolution 0 We’ll first look at the area between the lines y = x,... Ans: A cone ( lying on its side ) Can you see what shape you will get if you rotate the area through about the x -axis? x = 1,... and the x -axis. 1

4. Volumes of Revolution 0 1 r h For this cone, We’ll first look at the area between the lines y = x,... x = 1,... and the x -axis.

5. Volumes of Revolution x The formula for the volume found by rotating any area about the x -axis is a and b are the x -coordinates at the left- and right- hand edges of the area. where is the curve forming the upper edge of the area being rotated. a b We leave the answers in terms of

6. Volumes of Revolution r h 01 So, for our cone, using integration, we get We must substitute for y using before we integrate. I’ll outline the proof of the formula for you.

7. Volumes of Revolution x The formula can be proved by splitting the area into narrow strips... Each tiny piece is approximately a cylinder ( think of a penny on its side ). which are rotated about the x -axis. Each piece, or element, has a volume

8. Volumes of Revolution The formula can be proved by splitting the area into narrow strips... Each tiny piece is approximately a cylinder ( think of a penny on its side ). x Each piece, or element, has a volume which are rotated about the x -axis.

9. Volumes of Revolution The formula can be proved by splitting the area into narrow strips... Each tiny piece is approximately a cylinder ( think of a penny on its side ). x Each piece, or element, has a volume The formula comes from adding an infinite number of these elements. which are rotated about the x -axis.

10. Volumes of Revolution Solution: To find a volume we don’t need a sketch unless we are not sure what limits of integration we need. However, a sketch is often helpful. As these are the first examples I’ll sketch the curves. e.g. 1(a) The area formed by the curve and the x -axis from x = 0 to x = 1 is rotated through radians about the x - axis. Find the volume of the solid formed. (b) The area formed by the curve, the x -axis and the lines x = 0 and x = 2 is rotated through radians about the x - axis. Find the volume of the solid formed.

11. Volumes of Revolution area rotate about the x -axis A common error in finding a volume is to get wrong. So beware! (a) rotate the area between

12. Volumes of Revolution a = 0, b = 1 (a) rotate the area between

13. Volumes of Revolution

14. (b) Rotate the area between and the lines x = 0 and x = 2.

15. Volumes of Revolution (b) Rotate the area between and the lines x = 0 and x = 2.

16. Volumes of Revolution Remember that

17. Volumes of Revolution Exercise radians about the x -axis. Find the volume of the solid formed. 1(a) The area formed by the curve the x -axis and the lines x = 1 to x = 2 is rotated through (b) The area formed by the curve, the x -axis and the lines x = 0 and x = 2 is rotated through radians about the x - axis. Find the volume of the solid formed.

18. Volumes of Revolution Solutions: 1. (a), the x -axis and the lines x = 1 and x = 2.

19. Volumes of Revolution Solutions:

20. Volumes of Revolution Solution: (b), the x -axis and the lines x = 0 and x = 2.

21. Volumes of Revolution Students taking the EDEXCEL spec do not need to do the next ( final ) section.

22. Volumes of Revolution To rotate an area about the y -axis we use the same formula but with x and y swapped. The limits of integration are now values of y giving the top and bottom of the area that is rotated. Rotation about the y -axis As we have to substitute for x from the equation of the curve we will have to rearrange the equation. Tip: dx for rotating about the x -axis; dy for rotating about the y -axis.

23. Volumes of Revolution e.g. The area bounded by the curve, the y -axis and the line y = 2 is rotated through about the y -axis. Find the volume of the solid formed.

24. Volumes of Revolution

25. Exercise the y -axis and the line y = 3 is rotated through radians about the y -axis. Find the volume of the solid formed. 1(a) The area formed by the curve for (b) The area formed by the curve, the y -axis and the lines y = 1 and y = 2 is rotated through radians about the y -axis. Find the volume of the solid formed.

26. Volumes of Revolution Solutions: (a) for, the y -axis and the line y = 3.

27. Volumes of Revolution Solution: (b), the y -axis and the lines y = 1 and y = 2.

28. Volumes of Revolution

29. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

30. Volumes of Revolution x The formula for the volume found by rotating any area about the x -axis is a and b are the x -coordinates at the left- and right- hand edges of the area. where is the curve forming the upper edge of the area being rotated. a b We leave the answers in terms of

31. Volumes of Revolution e.g. 1 Find the volume of the solid formed by rotating through about the x -axis the area bounded by the given curves and lines. (b), the x -axis, and the lines x = 0 and x = 2. Solution: To find a volume we don’t need a sketch unless we aren’t sure what limits of integration we need. However, a sketch is often helpful. (a) and the x -axis from x = 0 to x = 1.

32. Volumes of Revolution area rotate about the x -axis A common error in finding a volume is to get wrong. So beware! (a) rotate the area between a = 0, b = 1

33. Volumes of Revolution

34. (b) Rotate the area between and the lines x = 0 and x = 2.

35. Volumes of Revolution Remember that

36. Volumes of Revolution STUDENTS TAKING THE EDEXCEL SPEC DO NOT NEED THIS SECTION. To rotate an area about the y -axis we use the same formula but with x and y swapped. The limits of integration are now values of y giving the top and bottom of the area that is rotated. Rotation about the y -axis As we have to substitute for x from the equation of the curve we will have to rearrange the equation.

37. Volumes of Revolution e.g. The area bounded by the curve, the y -axis and the line y = 2 is rotated through about the y -axis. Find the volume of the solid formed.

38. Volumes of Revolution