1. Integration
Volumes of revolution

2. FM Volumes of revolution II: modelling
KUS objectives
BAT Find Volumes of revolution using Integration
Starter: Find these integrals
𝑥𝑥𝑥𝐶
𝑥𝑥𝑥𝑥
𝑥𝑥𝑥𝑑𝑥
𝑥𝑥𝑥

3. 𝑥 =2 sin 𝑡 , and 𝑦 =2 cos 𝑡 + 2, 𝜋 6 ≤𝑡≤ 11𝜋 6
WB D1 The diagram shows a model of a goldfish bowl. The cross-section of the model is described by the curve with parametric equations
𝑥 =2 sin 𝑡 , and 𝑦 =2 cos 𝑡 + 2, 𝜋 6 ≤𝑡≤ 11𝜋 6
Where the units of x and y are given in cm. The goldfish bowl volume is formed by rotating the curve around the y-axis to form a solid of revolution.
a) Find the volume of the water required to fill the model to a height of 3 cm
b) The real goldfish bowl has a maximum diameter of 48 cm. Find the volume of water needed to fill the real bowl to a corresponding height.
4 cm
3 cm
𝑎) 𝑑𝑦 𝑑𝑡 =−2 sin 𝑡
𝑦=0 →𝑡=𝜋
𝑦=3 →𝑡= 𝜋 3 , 5𝜋 3 ,
Two values correspond to the two sides of the bowl
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝜋 5𝜋/ sin 𝑡 2 −2 sin 𝑡 𝑑𝑥
=−8𝜋 𝜋 5𝜋/3 𝑠𝑖𝑛 3 𝑡 𝑑𝑥
=−8𝜋 𝜋 5𝜋/3 sin 𝑡 (1− 𝑐𝑜𝑠 2 𝑡) 𝑑𝑥
=−8𝜋 𝜋 5𝜋/3 sin 𝑡 − sin 𝑡 𝑐𝑜𝑠 2 𝑡 𝑑𝑥

4. WB D1 (part 2) 𝑥 =2 sin 𝑡 , and 𝑦 =2 cos 𝑡 + 2, 𝜋 6 ≤𝑡≤ 11𝜋 6
a) Find the volume of the water required to fill the model to a height of 3 cm
b) The real goldfish bowl has a maximum diameter of 48 cm. Find the volume of water needed to fill the real bowl to a corresponding height.
=−8𝜋 𝜋 5𝜋/3 sin 𝑡 − sin 𝑡 𝑐𝑜𝑠 2 𝑡 𝑑𝑥
4 cm
3 cm
=−8𝜋 − cos 𝑡 − 1 3 𝑐𝑜𝑠 3 𝑡 5𝜋/3 𝜋
=−8𝜋 − cos 5𝜋 3 − 1 3 𝑐𝑜𝑠 3 5𝜋 3 + cos 𝜋 + 𝑐𝑜𝑠 3 𝜋
=…=9𝜋
𝑏) 𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟= 48 4 =12
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑤𝑙=1728× 9𝜋
= 𝑐𝑚 3
𝑎𝑟𝑒𝑎 𝑓𝑎𝑐𝑡𝑜𝑟= 12 2 =144
𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑎𝑐𝑡𝑜𝑟= 12 3 =1728

5. NOW DO Ex 4D 𝑦=− 1+ sin 𝑡 to get sin 𝑡 =−𝑦−1
WB D The diagram shows the image of a gold pendant which has height 2 cm. the pendant is modelled by a solid of revolution of a curve C about the y-axis. The curve has parametric equations 𝑥= cos 𝑡 sin 2𝑡 and 𝑦=− 1+ sin 𝑡 , 0≤𝑡≤2𝜋
a) Show that a Cartesian equation of the curve C is 𝑥 2 =− 𝑦 4 +2 𝑦 (4)
b) Hence, using the model, find in 𝑐𝑚 3 , the volume of the pendant (4)
First Rearrange
𝑦=− 1+ sin 𝑡 to get sin 𝑡 =−𝑦−1
𝑥= cos 𝑡 + sin 𝑡 cos 𝑡
= cos 𝑡 1+ sin 𝑡 =−𝑦 cos 𝑡
𝑥 2 = 𝑦 2 𝑐𝑜𝑠 2 𝑡 = 𝑦 2 1− 𝑠𝑖𝑛 2 𝑡
𝑥 2 = 𝑦 2 1− −𝑦−1 2
= 𝑦 −𝑦 2 −2𝑦
=− 𝑦 4 +2 𝑦 QED
b) volume =𝜋 −2 0 𝑥 2 𝑑𝑥
=𝜋 − 𝑦 4 +2 𝑦 𝑑𝑥
=1.6𝜋 𝑐𝑚 3
NOW DO Ex 4D

6. One thing to improve is –
KUS objectives BAT Find Volumes of revolution using Integration
self-assess
One thing learned is –
One thing to improve is –

7. END