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Volumes of solid (slicing, disc, washer, cylindrical shell)
- 1. “The goal in life is to be solid, whereas the way that life works is totally fluid, so you can never actually achieve that goal.” ~Damien Hirst by: Maria Faith L. Dalay
- 2. Let “S” be a solid that lies between x = a and x = b, if the cross-sectional area of “S” in the plane P(x) through x and perpendicular to the axis is A(x) then, 𝑉 = න 𝑎 𝑏 𝐴 𝑥 𝑑𝑥
- 4. Methods of finding the volume 1.Disc Method 2.Washer Method 3.Cylindrical/Shell Method 3
- 5. This method is useful when the axis of rotation is part of the boundary of the plane area.
- 6. Useful when the axis of rotation is not a part of the boundary of the plane area
- 7. Uses cylinder to compute the volume
- 8. Find the volume of the solid obtained by rotating about the x−axis the region under the curve y = √x from 0 to 1.
- 9. Find the volume of the solid obtained by rotating about the x−axis the region under the curve y = √x from 0 to 1. Radius = Top - Bottom = √x - 0 = √x Area of a circle = πr2
- 10. Find the volume of the solid obtained by rotating about the x−axis the region under the curve y = √x from 0 to 1. Radius = Top - Bottom = √x - 0 = √x 𝑉 = න 𝑎 𝑏 𝐴 𝑥 𝑑𝑥 𝑉 = න 𝑎 𝑏 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒 𝑑𝑥 𝑉 = න 𝑎 𝑏 π𝑟2 𝑑𝑥 𝑉 = න 0 1 π( 𝑥)2 𝑑𝑥 𝑉 = π න 0 1 𝑥 𝑑𝑥 𝑉 = π 𝑥2 2 1 0 𝑉 = π 12 2 − 0 𝑉 = π 2 unit 3
- 11. Find the volume of the solid obtained by rotating the region bounded by y = x3, y=8, y=0 about the y-axis.
- 12. Find the volume of the solid obtained by rotating the region bounded by y = x3, y=8, y=0 about the y-axis. y = x3 → x = ∛y Radius = Right - Left = ∛y - 0 = ∛y
- 13. Find the volume of the solid obtained by rotating the region bounded by y = x3, y=8, y=0 about the y-axis. y = x3 → x = ∛y Radius = Right – Left = ∛y – 0 = ∛y 𝑉 = න 𝑎 𝑏 (𝐴𝑟𝑒𝑎) 𝑑𝑦 𝑉 = න 𝑎 𝑏 π𝑟2 𝑑𝑦 𝑉 = න 0 8 π(∛y)2 𝑑𝑦 𝑉 = π න 0 8 (𝑦 2 3) 𝑑𝑦 𝑉 = π [ 3 5 𝑦 5 3] 8 0 𝑉 = 96π 5 unit 3
- 14. 𝑉 = π න 𝑎 𝑏 𝑟2 𝑑𝑥
- 15. 𝑉 = π න 𝑎 𝑏 (𝑟𝑜 𝑢𝑡)2 − (𝑟𝑖𝑛)2 𝑑𝑥
- 16. The region by the curves y = x and y = x2, is rotated about the x-axis. Find the volume of the rotating solid
- 17. The region by the curves y = x and y = x2, is rotated about the x-axis. Find the volume of the rotating solid Rout = Top - Bottom = x - 0 = x Rin = Top - Bottom = x2 - 0 = x2
- 18. The region by the curves y = x and y = x2, is rotated about the x-axis. Find the volume of the rotating solid Rout = Top - Bottom = x - 0 = x Rin = Top - Bottom = x2 - 0 = x2 𝑉 = π න 𝑎 𝑏 (𝑟𝑜 𝑢𝑡)2 − (𝑟𝑖𝑛)2 𝑑𝑥 𝑉 = π න 0 1 (𝑥)2 − (𝑥2)2 𝑑𝑥 𝑉 = π න 0 1 𝑥2 − 𝑥4 𝑑𝑥 𝑉 = π 𝑥3 3 − 𝑥5 5 1 0 𝑉 = π 𝑥3 3 − 𝑥5 5 1 0 𝑉 = π 1 3 − 1 5 − 0 𝑉 = π 2 15 𝑉 = 2π 15 unit 3
- 19. 𝑉 = 2π න 𝑎 𝑏 𝑥𝑦 𝑑𝑥 𝑉 = 2π න 𝑎 𝑏 𝑥𝑦 𝑑𝑦
- 20. Find the volume of the solid obtained by rotating about the x−axis the region under the curve y = √x from 0 to 1.
- 21. Find the volume of the solid obtained by rotating about the x−axis the region under the curve y = √x from 0 to 1. 𝑉 = 2π න 𝑎 𝑏 𝑥𝑦 𝑑𝑥 𝑉 = 2π න 𝑎 𝑏 𝑥𝑦 𝑑𝑦 radius height y 1 y2 1-y2 y = x → x =y2 Radius : y Height: 1-y2
- 22. π 2 unit 3 Find the volume of the solid obtained by rotating about the x−axis the region under the curve y = √x from 0 to 1. Radius : y Height : 1-y2 𝑉 = 2π න 𝑎 𝑏 𝑥𝑦 𝑑𝑦 𝑉 = 2π න 𝑎 𝑏 𝑥𝑦 𝑑𝑦 𝑉 = 2π න 0 1 (𝑦)(1 − 𝑦2)𝑑𝑥 𝑉 = 2π න 0 1 (𝑦 − 𝑦3)𝑑𝑥 𝑉 = 2π 1 2 𝑦2 − 1 4 𝑦4 1 0 𝑉 = 2π 1 2 𝑦2 − 1 4 𝑦4 1 0 𝑉 = 2π 1 4 − 0 𝑉 =
- 23. Find the volume of the solid obtained by rotating about the x−axis the region under the curve y = √x from 0 to 1. Radius = Top - Bottom = √x - 0 = √x 𝑉 = න 𝑎 𝑏 𝐴 𝑥 𝑑𝑥 𝑉 = 𝑉 = න 𝑎 𝑏 π𝑟2 𝑑𝑥 𝑉 = න 0 1 π( 𝑥)2 𝑑𝑥 𝑉 = π න 0 1 𝑥 𝑑𝑥 𝑉 = π 𝑥2 2 1 0 𝑉 = π 12 2 − 0 𝑉 = π 2 unit 3
- 24. The region by the curves y = x and y = x2, is rotated about the x-axis. Find the volume of the rotating solid
- 25. The region by the curves y = x and y = x2, is rotated about the x-axis. Find the volume of the rotating solid 𝑉 = 2π න 𝑎 𝑏 𝑥𝑦 𝑑𝑦 radius height y = x → x = y y = x2 → x =√y y Radius: y Height: √y − y y √y Height = Right - Left = √y − y
- 26. The region by the curves y = x and y = x2, is rotated about the x-axis. Find the volume of the rotating solid 𝑉 = 2π න 𝑎 𝑏 𝑥𝑦 𝑑𝑦 𝑉 = 2π න 0 1 (𝑦)(√𝑦 − 𝑦)𝑑𝑦 𝑉 = 2π න 0 1 (𝑦 3 2 − 𝑦2)𝑑𝑦 𝑉 = 2π 2 5 𝑦 5 2 − 1 3 𝑦3 1 0 2π 15 unit 3 𝑉 = 2π 2 5 𝑦 5 2 − 1 3 𝑦3 1 0 𝑉 = 2π 2 5 − 1 3 − 0 𝑉 = 2π 1 15 𝑉 = Radius: y Height: √y − y
- 27. The region by the curves y = x and y = x2, is rotated about the x-axis. Find the volume of the rotating solid Rout = Top - Bottom = x - 0 = x Rin = Top - Bottom = x2 - 0 = x2 𝑉 = π න 𝑎 𝑏 (𝑟𝑜 𝑢𝑡)2 − (𝑟𝑖𝑛)2 𝑑𝑥 𝑉 = π න 0 1 (𝑥)2 − (𝑥2)2 𝑑𝑥 𝑉 = π න 0 1 𝑥2 − 𝑥4 𝑑𝑥 𝑉 = π 𝑥3 3 − 𝑥5 5 1 0 𝑉 = π 𝑥3 3 − 𝑥5 5 1 0 𝑉 = π 1 3 − 1 5 − 0 𝑉 = π 2 15 𝑉 = 2π 15 unit 3
- 28. by: Maria Faith L. Dalay
- 29. Rotate the region bounded by y=√x, y=3 and the y-axis about the y-axis. Rotate the region bounded by x = y2 – 6y +10 and x = 5 about the y-axis. 1 2 Exercises: