11X1 T14 05 volumes

Download 11X1 T14 05 volumes

Post on 30-Jun-2015

342 views

Category:

Education

0 download

TRANSCRIPT

1. Volumes of Solids of Revolutiony y = f(x) x 2. Volumes of Solids of Revolutionyy = f(x)a b x 3. Volumes of Solids of Revolutionyy = f(x)a b x 4. Volumes of Solids of Revolutionyy = f(x)a b xVolume of a solid of revolution about; 5. Volumes of Solids of Revolution yy = f(x) a bxVolume of a solid of revolution about; i x axis : V y 2 dx ba 6. Volumes of Solids of Revolution y y = f(x) a bxVolume of a solid of revolution about; i x axis : V y 2 dxb aii y axis : V c x 2 dyd 7. e.g. (i) coney y mxh x 8. e.g. (i) coney y mxh x 9. e.g. (i) coneV y 2 dx hy m 2 x 2 dx y mx0 h x 10. e.g. (i) coneV y 2 dx hy m 2 x 2 dx y mx0 h 2 1 3 m x 3 0h x 11. e.g. (i) coneV y 2 dx hy m 2 x 2 dx y mx0 h 2 1 3 m x 3 0 m 2 h 3 0 1 h x 3 1 m 2 h 3 3 12. e.g. (i) coneV 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 13. e.g. (i) cone V 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 33r mh 14. e.g. (i) cone V 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 332r 1 r m h3h 3 hr 2 h 3 3h 2r 2 h3 15. (ii) spherey x2 y2 r 2rx 16. (ii) spherey x2 y2 r 2rx 17. (ii) spherey V y 2 dx x2 y2 r 2 r 2 r 2 x 2 dx 0 rx 18. (ii) spherey V y 2 dx x2 y2 r 2 r 2 r 2 x 2 dx 0r r 2 x 1 x 3 2 rx 3 0 19. (ii) spherey V y 2 dx x2 y2 r 2 r 2 r 2 x 2 dx 0r r 2 x 1 x 3 2 rx 3 0 2 r 2 r r 3 01 3 2 3 2 r 3 4 3 r3 20. (iii) Find the volume of the solid generated when y x 2 is revolved around the y axis between y 0 and y 1. 21. (iii) Find the volume of the solid generated when y x 2 is revolved around the y axis between y 0 and y 1.y y x2x 22. (iii) Find the volume of the solid generated when y x 2 is revolved around the y axis between y 0 and y 1.y y x21x 23. (iii) Find the volume of the solid generated when y x 2 is revolved around the y axis between y 0 and y 1.y y x21x 24. (iii) Find the volume of the solid generated when y x 2 is revolved around the y axis between y 0 and y 1.y y x2V x 2 dy1 1 ydy0x 25. (iii) Find the volume of the solid generated when y x 2 is revolved around the y axis between y 0 and y 1.y y x2V x 2 dy 1 1 ydy 0 2 y 2 10 x 26. (iii) Find the volume of the solid generated when y x 2 is revolved around the y axis between y 0 and y 1.y y x2V x 2 dy 1 1 ydy 0 2 y 2 1 0 x 1 2 0 2 units32 27. (iv) Find the volume of the solid when the shaded region is rotatedabout the x axis y y 5x 5 x 28. (iv) Find the volume of the solid when the shaded region is rotatedabout the x axis yV y 2 dx y 5x 1 5 52 5 x dx2 01 x 25 25 x 2 dx0 29. (iv) Find the volume of the solid when the shaded region is rotatedabout the x axis yV y 2 dx y 5x 1 5 52 5 x dx2 01 x 25 25 x 2 dx01 x 1 x3 25 3 0 30. (iv) Find the volume of the solid when the shaded region is rotatedabout the x axis yV y 2 dx y 5x 1 5 52 5 x dx2 01 x 25 25 x 2 dx01 x 1 x3 25 3 0 1 1 13 0 25 3 50 unit 33 31. (iv) Find the volume of the solid when the shaded region is rotatedabout the x axis yV y 2 dx y 5x 1 5 52 5 x dx2 01 x 25 25 x 2 dx01 x 1 x3 25 3 0 1 1 13 0OR 25 3 50 unit 33 32. (iv) Find the volume of the solid when the shaded region is rotatedabout the x axis yV y 2 dx y 5x 1 5 52 5 x dx2 01 x 25 25 x 2 dx01 x 1 x3 25 3 0 11 1 13 0OR V r 2 h r 2 h 25 3 3 2 50 5 1 2unit 3 33 50 units 3 3 33. 2005 HSC Question 6c) The graphs of the curves y x 2 and y 12 2 x 2 are shown in the diagram. (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 the diagram. (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 the diagram. (i) Find the points of intersection of the two curves. (1) x 2 12 2 x 2 3 x 2 12x2 4 x 2 36. 2005 HSC Question 6c) The graphs of the curves y x 2 and y 12 2 x 2 are shown in the diagram. (i) Find the points of intersection of the two curves. (1) x 2 12 2 x 2 3 x 2 12x2 4 x 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 twoparts, 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 twoparts, 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 twoparts, 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 twoparts, 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 twoparts, or otherwise, find the volume of the solid formed. x2 y V 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 twoparts, or otherwise, find the volume of the solid formed. x2 y 1 x 6 y2V 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 twoparts, or otherwise, find the volume of the solid formed. x2 y 1 x 6 y2V x 2 dy2412 1 V 6 y dy ydy 02 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 twoparts, or otherwise, find the volume of the solid formed. x2 y 1 x 6 y2V x 2 dy2412 1 V 6y dy ydy 0 2 4 4 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 twoparts, or otherwise, find the volume of the solid formed. x2 y 1 x 6 y2V x 2 dy2412 1 V 6y dy ydy 0 2 4 4 12 6 y y y 1 2 1 2 4 0 2 41 2 6 4 4 0 12 4 4 222 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 twoparts, or otherwise, find the volume of the solid formed. x2 y 1 x 6 y2V x 2 dy2412 1 V 6y dy ydy 0 2 4 4 12 6 y y y 1 2 1 2 4 0 2 41 2 6 4 4 0 12 4 4 222 84 units3 47. Exercise 11G; 3bdef, 4eg, 5cd, 6d, 8ad, 12, 16a, 19 to 22, 24*, 25*