Maths M2 Disc method (13)

2013-06-01 3:02 am

If the region bounded by the curve x = 0.5 e^y, the line y = 7, the positive x-axis and the positive y-axis is revolved about the x-axis, find the volume of the solid of revolution by the disc method.
Thanks.

回答 (1)

Shu Fun

2013-06-01 3:56 am

✔ 最佳答案

x = 0.5e^y, y = ln(2x)
When y = 0, x = 0.5
When y = 7, x = 0.5e^7

Required volume
=∫(0, 0.5e^7) π*7^2 dx - ∫(0.5, 0.5e^7) π[y^2] dx
=π*49*0.5e^7 - π∫(0.5, 0.5e^7) ln(2x)^2 dx
=24.5e^7π - π[xln(2x)^2](0.5, 0.5e^7) + π∫(0.5, 0.5e^7) x d(ln(2x)^2)
=24.5e^7π - π[0.5e^7*7^2] + π∫(0.5, 0.5e^7) 2xln(2x)/x dx
=π∫(0.5, 0.5e^7) 2ln(2x) dx
=π[2xln(2x)](0.5, 0.5e^7) - π∫(0.5, 0.5e^7) 2x d(ln(2x))
=7e^7π - π∫(0.5, 0.5e^7) 2x/x dx
=7e^7π - 2π*(0.5e^7-0.5)
=6πe^7+π

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