Expert: Paul Klarreich - 2/7/2008

Question
I am taking Integral Calculus and have the following problem.

Find the Volume, V, of the solid formed by rotating the region bounded by the graphs of
y = (x)^(1/2) = 4 , y = 4, x = 0, x = 1
about the line y = 3.

I am having a hard time setting this problem up because it is rotating about the y = 3 line instead of the x-axis. Can you please help me with this?

Answer
Questioner:   Danielle
Category:  Calculus
Private:  No
 
Subject:  Volume by Integration
Question:  I am taking Integral Calculus and have the following problem.

Find the Volume, V, of the solid formed by rotating the region bounded by the graphs of
y = (x)^(1/2) = 4 , y = 4, x = 0, x = 1

>> ------------^^ ?????
You meant  y = x^/2 COMMA, didn't you?


about the line y = 3.

I am having a hard time setting this problem up because it is rotating about the y = 3 line instead of the x-axis. Can you please help me with this?

..........................
Hi, Danielle,

To get your typical 'piece' of the volume, you take a 'slice' of the volume.  You get it by rotating a slice of area.  That area is in between:

y = 3  [upper]
y = x^1/2  [lower]

and a typical slice is a disk that has:

radius = 3 - x^1/2
thickness = dx.

volume dV = pi r^2 h = pi (3 - x^1/2)^2 dx

and the slices go from x = 0 to x = 1.

{1
|   pi (3 - x^1/2)^2 dx
}0

{1
|   pi (9 - 6x^1/2 + x) dx =
}0

pi (9x - 6x^3/2/(3/2) + x^2/2)

 pi (9x - 4x^3/2 + x^2/2) from 0 to 1 =

 pi (9 - 4 + 1/2) = 11/2 pi