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Tricky Multiple integral word problem

by numberonenacho
Tags: integral, multiple, tricky, word
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numberonenacho
#1
Nov24-07, 02:52 PM
P: 12
So Ive been talking with some people about this problem, but I cant seem to find the answer, or even set it up!

"A region in space, when viewed from 3 different views, looks like a circle, looks like a square and looks like a triangle. Describe this object. And then use multiple integration to determine its volume"

So if you think of any two shapes its pretty easy. Such as square and triangle = pyramid, circle and square = cylinder and so on. But I couldnt figure out a shape for when you add the third shape. And then Figuring out the integral would be even more difficult.
What do you guys think about it?
=D
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bob1182006
#2
Nov24-07, 03:34 PM
P: 492
you almost had it with the circle+square=cylinder, what if you were to have only a square from the "top/bottom" and the sides a triangle?
numberonenacho
#3
Nov24-07, 03:43 PM
P: 12
Im still not getting it.. Pictures would help. >.<
So Square base and top. Sides triangle? Where would the circle go?

bob1182006
#4
Nov24-07, 03:45 PM
P: 492
Tricky Multiple integral word problem

it's originally a cylinder, 2r=height, so from 2 sides you'd have a square. and now cut off some part in order to get a triangle from the other 2 sides. So it looks like a wedge with a circular base and square top/bottom.
numberonenacho
#5
Nov24-07, 04:40 PM
P: 12
OH I think I get it now. So bascially, its 1/2 a cylinder right?

http://img77.imageshack.us/img77/8178/shapehi9.gif

So side view is square, bottom is circle
and other sides are triangles. Thanks!!

But how would I find the volume of that shape?
bob1182006
#6
Nov24-07, 04:47 PM
P: 492
hm..depends how you place the axis's on it, but I think you'll need to add 2 integrals due to the circle you'll have the top and bottom part of it.

and it might be better to cut the cylinder by 2 lines so looking at the triangle side you can cut it at the center of the circle and get 2 symmetric sides so you can just do 2xsome integral to find the volume.
numberonenacho
#7
Nov24-07, 04:51 PM
P: 12
Well if I cut the cylinder in half diagonally, couldn't I just take the volume of the cylinder and then divide it by two? When you say cut the cylinder with two lines what do you mean?

And I have no idea how to set up the integral. >.<
could you get me started please?
bob1182006
#8
Nov24-07, 04:54 PM
P: 492
o yea you could just do that.

Use cylindrical coordinates to setup your integral and then just multiply it by .5 to find the volume.
numberonenacho
#9
Nov24-07, 05:15 PM
P: 12
could you help me set it up? thats the part im really bad at. Many thanks for discussing this problem with me. =)
bob1182006
#10
Nov24-07, 05:19 PM
P: 492
well you have [tex]r,\theta,z[/tex] in cylindrical coordinates.

r-?
theta-?
z-?

z being the height and from what we've written you know what it is.
so just find the r and the z and you can setup the integral.
numberonenacho
#11
Nov24-07, 05:30 PM
P: 12
theta would be 0 - 2pi?
z would be the height soo lets say 0 - a
what would r be?
bob1182006
#12
Nov24-07, 05:33 PM
P: 492
r would be 0 to the circle's radius, say R, and the height of the cylinder can be described in terms of R. So you can have the square when looking at it from 2 sides.
numberonenacho
#13
Nov24-07, 05:41 PM
P: 12
what would the actual integrand be? [tex]rdrdz\theta[/tex]
r - 0-R
theta - 0-2pi
z - 0-R

?
bob1182006
#14
Nov24-07, 05:47 PM
P: 492
no the z=2R because you need the diameter of the circle to = the height in order to get the square when looking at it from that side.

and yes the integrand will just be rdrdzd(theta).
numberonenacho
#15
Nov24-07, 05:55 PM
P: 12
[tex] 2\pi (\int \int rdrdz) [/tex]
0-2R for first integral and 0-R for second one. how would I go from there? the rdr is confusing me a little.
bob1182006
#16
Nov24-07, 05:59 PM
P: 492
[tex]\int_0^{2\pi}d\theta\int_0^R rdr\int_0^{2R}dz[/tex]

the rdr just means integrate r w.r.t. r just like: [tex]\int x dx[/tex]
numberonenacho
#17
Nov24-07, 06:06 PM
P: 12
I got 2pi r^3 for final answer
after dividing by two. How does that sound?
bob1182006
#18
Nov24-07, 06:09 PM
P: 492
yep that's what I got.

the volume for a cylinder is pi r^2 *h. h=2r ->2pir^3 and then *.5 = pi r^3


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