1. 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

2. 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.

3. 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

4. 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.

5. 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

6. 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.

7. 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.

8. 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.

9. 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

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

11. Volumes of Revolution

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

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

14. Volumes of Revolution Remember that

15. 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.

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

17. Volumes of Revolution Solutions:

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

19. 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.

20. 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.

21. Volumes of Revolution

22. 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.

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

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

25. Volumes of Revolution