Recommended
Recommended
More Related Content
What's hot
What's hot (19)
Similar to Volumes of solid (slicing, disc, washer, cylindrical shell)
Similar to Volumes of solid (slicing, disc, washer, cylindrical shell) (20)
Recently uploaded
Recently uploaded (20)
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: