1. Integration
Volumes of revolution

2. FM Volumes of revolution I: around yAxis
KUS objectives
BAT Find Volumes of revolution using Integration; find volumes by rotating around the y-axis
Starter:

3. Notes
The volume of revolution formed when x=f(y) is rotated around the y-axis between the y- axis, 𝒚=𝒂 and 𝒚=𝒃 is given by
𝑉𝑜𝑙𝑢𝑚𝑒=𝜋 𝑎 𝑏 𝑥 2 𝑑𝑥
When you use this formula you are integrating with respect to y. So you may need to rearrange functions accordingly

4. WB B1 The region R is bounded by the curve 𝑦= 𝑥−1 , the y-axis and the horizontal lines 𝑦=1 and 𝑦=3
Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π R
rearrange 𝑦= 𝑥−1 to 𝑥= 𝑦 2 +1
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 1 3 𝑥 2 𝑑𝑦=𝜋 𝑦 𝑑𝑦
=𝜋 𝑦 4 +2 𝑦 𝑑𝑦
= 𝜋 𝑦 𝑦 3 +𝑦 3 1
= 𝜋 − = 𝜋

5. WB B2 The region R is bounded by the curve 𝑥= 2𝑦−1 , the y-axis and the vertical lines y=4 and y = 8
Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π
𝑥= 2𝑦−1
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑦− 𝑑𝑦
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑦−1 𝑑𝑦
= 𝜋 𝑦 2 −𝑦 8 4
= 𝜋 64−8 −𝜋 16−4 =44𝜋

6. WB B3
The region R is bounded by the curve 𝑦= 𝑥 2 −2 the y-axis and the vertical lines 𝑦=1 and 𝑦=3
Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π
rearrange 𝑦= 𝑥 2 −2 to x = (𝑦+2) 1/2
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 (𝑦+2) 1/ 𝑑𝑦
=𝜋 1 3 (2+𝑦) 𝑑𝑦
= 𝜋 2𝑦 𝑦
=𝜋 −2− = 8𝜋

7. WB B4
The region R is bounded by the curve 𝑦= 1 𝑥 the y-axis and the vertical lines 𝑦=1 and 𝑦=2
Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π
rearrange 𝑦= 1 𝑥 to x = 1 𝑦
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑦 𝑑𝑦
=𝜋 𝑦 2 𝑑𝑦
= 𝜋 − 1 𝑦 2 1
= 𝜋 2

8. WB B5 The region R is bounded by the curve 𝑦= 𝑥 the y-axis and the vertical line 𝑦=2
Find the volume of the solid formed when the region is rotated 2π radians about the y-axis.
rearrange 𝑥= 𝑦 2
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑦 𝑑𝑦
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 0 2 𝑦 4 𝑑𝑦
= 𝜋 𝑦
= 32𝜋 5
NOW DO Ex 5B

9. One thing to improve is –
KUS objectives
BAT Find Volumes of revolution using Integration; find volumes by rotating around the y-axis
self-assess
One thing learned is –
One thing to improve is –

10. END