1. Integration
Volumes of revolution

2. FM Volumes of revolution II: around y-axis
KUS objectives
BAT Find Volumes of revolution using Integration; Rotations around the yAxis
Starter: Find these integrals
𝑥𝑥𝑥𝐶
𝑥𝑥𝑥𝑥
𝑥𝑥𝑥𝑑𝑥
𝑥𝑥𝑥

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 =4 ln 𝑥 −1, the y-axis, x-axis and the horizontal lines y = 0 and y = 4
Show that the volume of the solid formed when the region is rotated 2π radians about the y-axis is 2𝜋 𝑒 𝑒 2 −1
𝑥= 𝑒 𝑦 = 𝑒 𝑒 𝑦 4
𝑦 =4 ln 𝑥 −1, rearranges to
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑒 𝑒 𝑦 𝑑𝑦
=𝜋 𝑒 𝑒 𝑦 2 𝑑𝑦
=𝜋 𝑒 𝑒 𝑦
=𝜋 𝑒 𝑒 2 − 𝑒 0
=2𝜋 𝑒 𝑒 2 − QED

5. WB B2
The region R is bounded by the curve 𝑦= ln 𝑥 the y-axis and the vertical lines 𝑦=1 and 𝑦=5
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= ln 𝑥 to x = 𝑒 𝑦
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑒 𝑦 𝑑𝑦
=𝜋 1 5 𝑒 2𝑦 𝑑𝑦
= 𝜋 𝑒 2𝑦 5 1
= 𝜋 2 𝑒 10 − 𝑒 2

6. WB B The area bounded by the curve y= 𝑥 2 and the lines 𝑥=3 and 𝑦=1 is rotated 2𝜋 about the line 𝑦=1
Find the volume of the solid formed
NOW DO Ex 4B
transform the graph by the shift f(x)-1
𝑦= 𝑥 2 becomes 𝑦= 𝑥 2 −1
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑥 2 − 𝑑𝑥
=𝜋 𝑥 4 −2 𝑥 𝑑𝑥 R
= 𝜋
Can you generalise to give a formula for the volume formed when the curve is rotated about line 𝑦=𝑎
transform the graph by the shift f(x)-a so 𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑎 𝑏 𝑓 𝑥 −𝑎 2 𝑑𝑥

7. One thing to improve is –
KUS objectives BAT Find Volumes of revolution using Integration; Rotations around the yAxis
self-assess
One thing learned is –
One thing to improve is –

8. END