# 11X1 T16 05 volumes (2011)

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• 1. Volumes of Solidsof Revolution yy = f(x)x

2. Volumes of Solidsof Revolution y y = f(x) a b x 3. Volumes of Solidsof Revolution y y = f(x) a b x 4. Volumes of Solidsof Revolution y y = f(x) a b xVolume of a solid of revolution about; 5. Volumes of Solidsof Revolutiony y = f(x)a bxVolume of a solid of revolution about; i x axis : V y 2 dxb a 6. Volumes of Solidsof Revolutionyy = f(x)a bxVolume of a solid of revolution about; i x axis : V y 2 dx b aii y axis : V c x 2 dy d 7. e.g. (i) coneyy mxh x 8. e.g. (i) coneyy mxh x 9. e.g. (i) coneV y 2 dxhy m 2 x 2 dxy mx0h x 10. e.g. (i) coneV y 2 dxhy m 2 x 2 dxy mx0h2 1 3 m x 3 0h x 11. e.g. (i) coneV y 2 dxhy m 2 x 2 dxy mx0h2 1 3 m x 3 0 m 2 h 3 0 1h x 31 m 2 h 33 12. e.g. (i) coneV y 2 dxhy m 2 x 2 dxy mx0h2 1 3 m x 3 0r m 2 h 3 0 1h x 31 m 2 h 33 13. e.g. (i) cone V y 2 dx hy m 2 x 2 dx y mx0 h 2 1 3 m x 3 0 r m 2 h 3 0 1 h x 3 1 m 2 h 3 3 rm h 14. e.g. (i) cone V y 2 dx hy m 2 x 2 dx y mx0 h 2 1 3 m x 3 0 r m 2 h 3 0 1 h x 3 1 m 2 h 3 32 r h3m 1 r h 3 h r 2 h 3 3h 2 r 2 h 3 15. (ii) sphereyx2 y2 r 2rx 16. (ii) sphereyx2 y2 r 2rx 17. (ii) spherey V y 2 dxx2 y2 r 2 r 2 r 2 x 2 dx0rx 18. (ii) spherey V y 2 dxx2 y2 r 2 r 2 r 2 x 2 dx0 rr 2 x 1 x 3 2 rx3 0 19. (ii) spherey V y 2 dxx2 y2 r 2 r 2 r 2 x 2 dx0 rr 2 x 1 x 3 2 rx3 01 2 r 2 r r 3 0 3 2 3 2 r 3 4 3 r 3 20. (iii) Find the volume of the solid generated when y x 2 is revolvedaround the y axis between y 0 and y 1. 21. (iii) Find the volume of the solid generated when y x 2 is revolvedaround the y axis between y 0 and y 1.y y x2 x 22. (iii) Find the volume of the solid generated when y x 2 is revolvedaround the y axis between y 0 and y 1.y y x21 x 23. (iii) Find the volume of the solid generated when y x 2 is revolvedaround the y axis between y 0 and y 1.y y x21 x 24. (iii) Find the volume of the solid generated when y x 2 is revolvedaround the y axis between y 0 and y 1.y y x2V x 2 dy 11 ydy 0 x 25. (iii) Find the volume of the solid generated when y x 2 is revolvedaround the y axis between y 0 and y 1.y y x2V x 2 dy11 ydy02y 2 1 0 x 26. (iii) Find the volume of the solid generated when y x 2 is revolvedaround the y axis between y 0 and y 1.y y x2V x 2 dy11 ydy02y 2 10 x 1 2 02 units32 27. (iv) Find the volume of the solid when the shaded region is rotated about the x axisyy 5x5 x 28. (iv) Find the volume of the solid when the shaded region is rotated about the x axisyV y 2 dxy 5x 15 52 5 x dx 2 0 1 x 25 25 x 2 dx 0 29. (iv) Find the volume of the solid when the shaded region is rotated about the x axisyV y 2 dxy 5x 15 52 5 x dx 2 0 1 x 25 25 x 2 dx 0 1 x 1 x3 25 3 0 30. (iv) Find the volume of the solid when the shaded region is rotated about the x axisyV y 2 dxy 5x 15 52 5 x dx 2 0 1 x 25 25 x 2 dx 0 1 x 1 x3 25 3 0 1 1 13 0 25 350unit 3 3 31. (iv) Find the volume of the solid when the shaded region is rotated about the x axisyV y 2 dxy 5x 15 52 5 x dx 2 0 1 x 25 25 x 2 dx 0 1 x 1 x3 25 3 0 1 1 13 0 OR 25 350unit 3 3 32. (iv) Find the volume of the solid when the shaded region is rotated about the x axisyV y 2 dxy 5x 15 52 5 x dx 2 0 1 x 25 25 x 2 dx 0 1 x 1 x3 25 3 0 11 1 13 0 OR V r 2 h r 2 h 25 3 32 50 5 1 2 unit 33350 units 33 33. 2005 HSC Question 6c)The graphs of the curves y x 2 and y 12 2 x 2 are shown in thediagram.(i) Find the points of intersection of the two curves. (1) 34. 2005 HSC Question 6c)The graphs of the curves y x 2 and y 12 2 x 2 are shown in thediagram.(i) Find the points of intersection of the two curves. (1) x 2 12 2 x 2 35. 2005 HSC Question 6c)The graphs of the curves y x 2 and y 12 2 x 2 are shown in thediagram.(i) Find the points of intersection of the two curves. (1)x 2 12 2 x 23 x 2 12 x2 4x 2 36. 2005 HSC Question 6c)The graphs of the curves y x 2 and y 12 2 x 2 are shown in thediagram.(i) Find the points of intersection of the two curves. (1)x 2 12 2 x 23 x 2 12 x2 4x 2 meet at 2, 4 37. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. 38. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. 39. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed.V x 2 dy 40. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed.V x 2 dy 41. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2 yV x 2 dy 42. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2 y1x 6 y 2V x 2 dy2 43. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2 y1x 6 y 2V x 2 dy2 412 1 V 6 y dy ydy0 2 4 44. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2 y1x 6 y 2V x 2 dy2 412 1 V 6y dy ydy02 44 12 6 y y y 1 2 1 2 4 0 2 4 45. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2 y1x 6 y 2V x 2 dy2 412 1 V 6y dy ydy02 44 12 6 y y y 1 2 1 2 4 0 2 4 1 2 6 4 4 0 12 4 4 2 22 46. (ii) The shaded region between the two curves and the y axis is(3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2 y1x 6 y 2V x 2 dy2 412 1 V 6y dy ydy02 44 12 6 y y y 1 2 1 2 4 0 2 4 1 2 6 4 4 0 12 4 4 2 22 84 units3 47. Exercise 11G; 3bdef, 4eg, 5cd, 6d, 8ad,12, 16a, 19 to 22, 24*, 25*