Integration Volumes of revolution

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 = 1 2 1− cos 2𝑥 𝑑𝑥= 1 2 𝑥− 1 2 sin 2𝑥 +𝐶

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

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?

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 0 2 𝑒 𝑥 2 𝑑𝑥 = 𝜋 1 2 𝑒 2𝑥 2 0 = 𝜋 2 𝑒 4 −1

WB A2 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 0 4 𝑒 2𝑥 2 𝑑𝑥 𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 0 4 𝑒 4𝑥 𝑑𝑥 = 𝜋 1 4 𝑒 4𝑥 4 0 = 𝜋 1 4 𝑒 16 −𝜋 1 4 = 𝜋 4 𝑒 16 −1

WB A3 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 3 − 3 6 = 5 3 6 𝜋

WB A4 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 2 𝑑𝑥 𝑠𝑖𝑛 2 𝑥= 1 2 − 1 2 cos 2x 𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 0 𝜋/2 1 2 − 1 2 cos 4𝑥 𝑑𝑥 = 𝜋 1 2 𝑥− 1 8 sin 4𝑥 𝜋/2 0 = 𝜋 𝜋 4 −0 − 0 −0 = 1 4 𝜋 2 NOW DO Ex 4A

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 –

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