# rotation about the y-axis (1 Viewer)

#### crazyymonkey

##### New Member

The Interior of the bowl which is to be used to hold the Olympic flame is shaped by rotating the arc of the curve y= loge x from x= 1 to x= 5 about the y-axis.

(i) SHow that the volume is given by V= (pi) S e^2y dy
[with b= ln5 and a= 0 ]

if that makes sense.

(ii) claculate the capacity of the bowl

#### tommykins

##### i am number -e^i*pi
y = lnx
x = e^y

Rotation of y axis makes it pi int. x^2 dy

therefore the volume is pi int. e^2y dy

domains are found by subbing x = 5 and x = 1 to find y.

ii) v = pi int. e^2y dy
= pi[(e^2y)/2] sub in ln 5 and 0 and you have your answer.

Last edited:

#### Just.Snaz

##### Member
Rotation about the y-axis: make x the subject

Y = lnx

x = e^y

V = pi integrate (from a to b) [ x^2 ] dy

since you'll be integrating in terms of y, you need your limits to be y values

so when x = 1, y = ln1 = 0
x = 5, y = ln5

therefore,

V = pi integrate (from 0 to ln5) [ e^2y ] dy

= pi [ (1/2)e^2y](limits 0 to ln5)
= pi [ (1/2)e^2ln5 - (1/2)e^0)
= pi [ (1/2)x25 - 1/2)
= pi (12)
= 12 pi