Integration Volumes of revolution

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

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

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+1 4 = 𝑒 1 4 𝑒 𝑦 4 𝑦 =4 ln 𝑥 −1, rearranges to 𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 0 4 𝑒 1 4 𝑒 𝑦 4 2 𝑑𝑦 =𝜋 𝑒 1 2 0 4 𝑒 𝑦 2 𝑑𝑦 =𝜋 𝑒 1 2 2 𝑒 𝑦 2 4 0 =𝜋 𝑒 1 2 𝑒 2 − 𝑒 0 =2𝜋 𝑒 𝑒 2 −1 QED

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 1 5 𝑒 𝑦 2 𝑑𝑦 =𝜋 1 5 𝑒 2𝑦 𝑑𝑦 = 𝜋 1 2 𝑒 2𝑦 5 1 = 𝜋 2 𝑒 10 − 𝑒 2

WB B3 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 1 3 𝑥 2 −1 2 𝑑𝑥 =𝜋 1 3 𝑥 4 −2 𝑥 2 +1 𝑑𝑥 R = 496 15 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

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 –

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