The document discusses methods for finding the volume of solids obtained by rotating planar regions about axes. It provides three main methods - the disc method, washer method, and cylindrical/shell method - and examples of applying each method to find the volume of various solids.

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: