Volumes by slicing (1 Viewer)

[shinn]

I can do this question by the shells method, but I want to know how to do it by volumes by slicing/washer method.

Question:
A circle of radius 2 is given by x^2+y^2 = 4 and is revolved about the line x=5. Using the washer/volume by slicing method, prove that the resulting volume (a TORUS) is 40 pie^2.


Any help would be greatly appreciated!

P.s. this question is from the 4U patell book.

 

[uniform]

This is how i would do it; i'm not exactly sure if it's the most appropriate/efficient way, and it's hard for me to explain without a diagram, but here goes:

By rearranging the equation of the circle making x the subject, you have
x = +/-sqrt(4-y^2)

so the corresponding x values for the outer and inner radii for each washer (i.e. any two collinear points on the circle cut by a line parallel to the x axis) are
x = -sqrt(4-y^2) and x = +sqrt(4-y^2) respectively,
BUT since the circle is rotated around x=5, the outer and inner radii lengths become
[5 + sqrt(4-y^2)] and [5 - sqrt(4-y^2)] respectively, by observation of distances from x = 5 to the above x values, on a diagram

therefore the volume of each 'washer' is
delta.V = pi{[5 + sqrt(4-y^2)]^2 - [5 - sqrt(4-y^2)]^2}delta.y
(i.e. delta.V = pi{outerradius^2 - innerradius^2} * height)

which simplifies (using principle that (a^2 - b^2) = (a-b)(a+b)) to
delta.V = pi{10[2sqrt(4-y^2)]}delta.y
delta.V = pi{20sqrt(4-y^2)}delta.y

therefore Volume is found by integrating the above from y = 2 to y = -2 with respect to y,
V = 20pi * [I{-2 -> 2)sqrt(4-y^2)dy]
but the integral is the same expression as finding the area of a semicircle on its side from y = 2 to -2, so
V = 20pi * 1/2 * pi * 2^2, therefore
V = 40pi^2 as required

hope that helps

 

[nottellingu]

HSC 2008 and ur up to volumes already....whoa !!!

 

[Mark576]

He's probably working on volumes these holidays to get ahead, don't you think?

 

[shinn]

thx alot.... i never thought of spliting it into two semicircles D=

 

[wogblogger]

OMG u genious
haha we have only done numbers and polynomials

 

[nottellingu]

He's probably working on volumes these holidays to get ahead, don't you think?

Doesnt dat mean he's already finished integration as well.
Btw i admre the guy...not paying him out

 

[nottellingu]

We've done complex and curve sketching i guess conics is next or maybe polynomials....hmm maybe i should gat ahead