Question

How do you find the volume bounded by `y = 12 ln x`, the x-axis, the y-axis and the line y=12 ln14 revolved about the y-axis?

Answer 1

`1170 pi`

Explanation:

Working up the y axis, the elemental volume of a small disc of thickness `Delta y` revolved about the y axis will be `Delta V = \pi x^2 Delta y`

where: `x = e^{y/12}` because `y = 12 ln(x)` ..... and so `x^2 = e^{y/6}`

so `V = pi \ int_0^{12 ln(14)} \ e^{y/6} \ dy`
`= 6 \pi [ e^{y/6} ]_0^{12 ln(14)}`
`= 6 pi [ exp((12 ln(14))/6) - 1]`
`= 6 pi [ e^(2 ln(14)) - 1]`
`= 6 pi [ e^( ln(14^2)) - 1]`
`= 6 pi [ 14^2 - 1] = 1170 pi`