1. Integration
Volumes of revolution

2. FM Volumes of revolution II: around x-axis
KUS objectives
BAT Find Volumes of revolution using Integration; rotating around xAxis
Starter: Find these integrals
= 1 6 𝑒 6𝑥 +𝐶
𝑒 6𝑥 𝑥 𝑑𝑥
cos 5𝑥 𝑑𝑥
= 1 5 sin 5𝑥 +𝐶
𝑠𝑖𝑛 2 𝑥 𝑑𝑥
= − cos 2𝑥 𝑑𝑥= 1 2 𝑥− 1 2 sin 2𝑥 +𝐶

3. Notes 1
You can use Integration to find areas and volumes y y x a b dx x a b y dx
In the trapezium rule we thought of the area under a curve being split into trapezia.
To simplify this explanation, we will use rectangles now instead
The height of each rectangle is y at its x-coordinate
The width of each is dx, the change in x values
So the area beneath the curve is the sum of ydx (base x height)
The EXACT value is calculated by integrating y with respect to x (y dx)
For the volume of revolution, each rectangle in the area would become a ‘disc’, a cylinder
The radius of each cylinder would be equal to y
The height of each cylinder is dx, the change in x
So the volume of each cylinder would be given by πy2dx
The EXACT value is calculated by integrating y2 with respect to x, then multiplying by π. (πy2 dx)
𝑎𝑟𝑒𝑎= 𝑎 𝑏 𝑦 𝑑𝑥

4. Notes 2 𝑎𝑟𝑒𝑎= 𝑎 𝑏 𝑦 𝑑𝑥 𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜋 𝑎 𝑏 𝑦 2 𝑑𝑥 x y y x a b
This would be the solid formed
𝑎𝑟𝑒𝑎= 𝑎 𝑏 𝑦 𝑑𝑥
𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜋 𝑎 𝑏 𝑦 2 𝑑𝑥
Imagine we rotated the area shaded around the x-axis
What would be the shape of the solid formed?

5. WB A1 The region R is bounded by the curve 𝑦 = 𝑒 𝑥 , the x-axis and the vertical lines 𝑥=0 and 𝑥=2
Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give an exact answer
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑒 𝑥 𝑑𝑥
= 𝜋 𝑒 2𝑥 2 0
= 𝜋 𝑒 4 −1

6. WB A The region R is bounded by the curve 𝑦 = 𝑒 2𝑥 , the x-axis and the vertical lines x = 0 and x = 4. Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Write your answer as a multiple of π
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑒 2𝑥 𝑑𝑥
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑒 4𝑥 𝑑𝑥
= 𝜋 𝑒 4𝑥 4 0
= 𝜋 𝑒 16 −𝜋 = 𝜋 4 𝑒 16 −1

7. WB A The region R is bounded by the curve 𝑦 = sec 𝑥 , the x-axis and the vertical lines 𝑥= 𝜋 6 and 𝑥= 𝜋 3
Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give an exact answer
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝜋/6 𝜋/3 𝑠𝑒𝑐𝑥 2 𝑑𝑥
𝑑 𝑑𝑥 ( tan 𝑥 )= 𝑠𝑒𝑐 2 𝑥
= 𝜋 tan 𝑥 𝜋/3 𝜋/6
= 𝜋 − = 𝜋

8. WB A The region R is bounded by the curve 𝑦 = sin2𝑥, the x-axis and the vertical lines x = 0 and x = π/2
Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give your answer as a multiple of 𝜋 2
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 0 𝜋/2 sin 2𝑥 𝑑𝑥
𝑠𝑖𝑛 2 𝑥= 1 2 − 1 2 cos 2x
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 0 𝜋/ − 1 2 cos 4𝑥 𝑑𝑥
= 𝜋 𝑥− 1 8 sin 4𝑥 𝜋/2 0
= 𝜋 𝜋 4 −0 − 0 −0 = 𝜋 2
NOW DO Ex 4A

9. One thing to improve is –
KUS objectives
BAT Find Volumes of revolution using Integration; rotating around xAxis
self-assess
One thing learned is –
One thing to improve is –

10. END