Find the volume of a solid between y = 0, y = x and x^2 + z^2 = 1

[Ryker]

Homework Statement

Find the volume of a solid between y = 0, y = x and x² + z² = 1 for all y >= 0.

The Attempt at a Solution

I'm having some trouble visualizing this, and what I'm not sure about is whether this is just a pyramid inside a cylinder (ie. straight lines from the base to the vertex, which I assume is at (1, 1, 0)) or whether the the volume of this solid is actually a sum of circular segments, so that the lines curve a bit. If the former is true, then finding the volume isn't hard, as you know the formula, but if the latter is true, then any suggestions on how to continue would be greatly appreciated.

Thanks in advance.

edit: After thinking about it some more, I think this is not a pyramid, and the lines to the vertex aren't straight (please correct me if I'm wrong). Now it seems I am in a pickle

So with this in mind, I tried drawing the view of this solid from the back, so that every slice looks like a circular segment. Then to get the volume I would find the volume of a solid with the slice area of a narrowing circular sector, and then subtract from that the volume of a solid with the slice area of a narrowing "circular triangle" (ie. sector - segment). Does this make sense at all or should I explain my line of logic in more detail?

If it does make sense, then since the radius of the circle is 1, all integrals would be from 0 to 1, and the final integral would be:

V = \int^1_0 arccosxdx - \int^1_0 x\sqrt{1-x^2}dx

Then I'd need to solve for this, but before I do that, I just wanted to see if this makes sense. It's solvable, so I'm happy with that, since when I first considered the option of this not being a pyramid, everything just started spinning (), but I'm not exactly sure I did it properly.

 

[the_kool_guy]

is answer 2/3

 

[tiny-tim]

Hi Ryker!

So with this in mind, I tried drawing the view of this solid from the back, so that every slice looks like a circular segment. Then to get the volume I would find the volume of a solid with the slice area of a narrowing circular sector, and then subtract from that the volume of a solid with the slice area of a narrowing "circular triangle" (ie. sector - segment). Does this make sense at all or should I explain my line of logic in more detail?

I'm not following that.

There's a cylinder along the y axis, cut by two vertical planes, one through the x axis, and one through y = x.

 

[Ryker]

is answer 2/3

Hi Ryker!
I'm not following that.

There's a cylinder along the y axis, cut by two vertical planes, one through the x axis, and one through y = x.

Thanks for the replies, both of you. Is the above answer, 2/3, correct? Because after correcting my integral above, this is the answer I get, if I follow the integral I set up. And as far as visualizing, I think I do get the same picture you are suggesting tiny-tim So if you would also happen to know the answer, that is, whether it's 2/3, it would be great if you could share it so that I can see whether I'm going wrong somewhere

 

[the_kool_guy]

i did it quite a simple way...
integration dydxdz
y (0,x)
x ( 0,(1 - z^2)^1/2 )
z ( -1,1 )...

i think its a solid whose projection in xz plane in semi circle...

 

[Ryker]

I'm not following that.

OK, I'll try to explain it a bit better. Suppose you're centered at the origin and you're looking at the cylinder from a perspective, such that you see its cross-section, that is, the circle. Now imagine the y = x plane cutting through the cylinder and position yourself in a way such that you stand at the beginning of this slope and you see the plane rising in front of you. Then at the very bottom this slope intersects the circle in the middle, a bit further up it intersects the circle a bit above the middle etc., until finally you reach the top. Each slice is thus the circular segment above the line where the y = x plane cuts the circle (inside the cylinder).

I don't know, to me it makes sense, but I can see why you're confused by what I'm saying as it's kind of hard to explain I guess. Still, if you'd happen to know the answer, and whether it's 2/3 then that'd be swell, as I could then check whether my reasoning is correct But if not, I'll need to go about it in another way. Any suggestions (provided the answer is wrong, that is )?

i did it quite a simple way...
integration dydxdz
y (0,x)
x ( 0,(1 - z^2)^1/2 )
z ( -1,1 )...

i think its a solid whose projection in xz plane in semi circle...

Hmm, we haven't done triple integrals yet, and I'm not really sure what course deals with that (the third course in Calculus perhaps?). So I can't really follow what you did there. But as for the projection, I don't really think it's a semi-circle, but rather, as I mentioned, a circular segment. Or perhaps I don't quite get what you're saying. Are you saying each slice is a semi-circle?

 

[the_kool_guy]

when you cut the cylinder x^2 + z^2 = 1 with a inclined plane x=y,the real projection of cut surface is an ellipse.
its just that the ellipse is being cut from middle through its minor axis axis... y=0
the ellipse i am talking about is in x=y plane...
...
.
your approach is quite different and unique i have heard of for evaluating volume.could you elaborate on it or give a link relating to that method... :)

 

[tiny-tim]

Suppose you're centered at the origin and you're looking at the cylinder from a perspective, such that you see its cross-section, that is, the circle. Now imagine the y = x plane cutting through the cylinder and position yourself in a way such that you stand at the beginning of this slope and you see the plane rising in front of you. Then at the very bottom this slope intersects the circle in the middle, a bit further up it intersects the circle a bit above the middle etc., until finally you reach the top. Each slice is thus the circular segment above the line where the y = x plane cuts the circle (inside the cylinder).

I'm still not following.

Imagine the cylinder is standing on its end (ie y is up) …

then the two planes y =0 amd y = x are the base of the cylinder, and a 45° slope through the centre of the base.

 

[Ryker]

OK, an update. It seems what I've been doing was correct tiny-tim, what method did you have in mind, though? Triangles or rectangles?

 

[tiny-tim]

I'd have used slices perpendicular to the y-axis …

that's triangles and bits of a circle.

 

[Ryker]

Hmm, what do you mean by bits of a circle? The triangle approach should namely work without considering anything else than triangles.

 

[tiny-tim]

Hmm, what do you mean by bits of a circle?

The bits near the circular edge of the cylinder.

 

[Ryker]

But you cover those with triangles, as well. Heh, you don't see my solution, I don't see yours, fun times