Integration Volumes of revolution

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

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/3 2 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

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 =48900 𝑐𝑚 3 𝑎𝑟𝑒𝑎 𝑓𝑎𝑐𝑡𝑜𝑟= 12 2 =144 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑎𝑐𝑡𝑜𝑟= 12 3 =1728

NOW DO Ex 4D 𝑦=− 1+ sin 𝑡 to get sin 𝑡 =−𝑦−1 WB D2 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 + 1 2 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 3 (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 2 −𝑦 2 −2𝑦 =− 𝑦 4 +2 𝑦 3 QED b) volume =𝜋 −2 0 𝑥 2 𝑑𝑥 =𝜋 −2 0 𝑦 4 +2 𝑦 3 2 𝑑𝑥 =1.6𝜋 𝑐𝑚 3 NOW DO Ex 4D

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 –

END