Solving for Volume of 3-D Structures

[Saitama]

Homework Statement

(see attachment)

Homework Equations

The Attempt at a Solution

How the equation $x^2+y^2-x=0$ is that of a cylinder? That looks to me an equation of a circle. I haven't learned about the different 3-D structures so I tried solving this using graphs. I tried to find the vertex of the paraboloid, it came out to be (0,0,1). Is this right? I am not sure how to proceed further from here.

Any help is appreciated. Thanks!

 

[voko]

How the equation $x^2+y^2-x=0$ is that of a cylinder? That looks to me an equation of a circle.

You get the cylinder by moving the circle along the z-axis. You can move the circle because z is not fixed by equation.

I haven't learned about the different 3-D structures so I tried solving this using graphs. I tried to find the vertex of the paraboloid, it came out to be (0,0,1). Is this right? I am not sure how to proceed further from here.

You should definitely sketch the problem, in order to figure out the integration domain and the integrand.

 

[jedishrfu]

for the cylinder the axis is along the z-axis if the center is at x=0 y=0 then x^2 + y^2 = R^2 in a plane would be a circle but along z is a cylinder.

Now, if the center of the cylinder is not at x but say at h then the equation would be rewritten as:

(x-h)^2 + y^2 = R^2 and expanding the (x-h)^2 term you'd get x^2 -2hx + h^2

and inserting that back into the equation gives you something like your equation x^2 + y^2 - x = 0
so its a cylinder along the z-axis but not with a center at x=0 and y=0

 

[Saitama]

Thanks both of you but I am having trouble visualising the figure. I drew a sketch but I am unable to find the volume cut out by the parabola. The cylinder extends infinitely aand at some point cuts the parabola. I can't form the integral for finding out the volume.

 

[voko]

Let's start with the domain first. One of the limiting planes is z = 0, so your domain could be in the XY plane. What does it look like?

 

[Saitama]

Two circles. One with centre at (1/2,0) and radius 1/2, the other is a circle of radius 1 centred at origin formed due to paraboloid.

 

[voko]

What about the x = 0 and y = 0 planes?

 

[Saitama]

What about the x = 0 and y = 0 planes?

In both the cases, a parabola with vertex at (0,0,1).

 

[voko]

I did not make myself clear. In the XY plane, what do the x = 0 and y = 0 conditions mean with regard to the domain? You have already found two "nested" circles there - but where exactly is the structure located?

 

[Saitama]

I did not make myself clear. In the XY plane, what do the x = 0 and y = 0 conditions mean with regard to the domain? You have already found two "nested" circles there - but where exactly is the structure located?

Haven't I already said that its a parabola rotated about the z-axis? I am not able to understand what you mean when you say where it is located [size= 1](and that's because of my poor English)[/size].

 

[voko]

Here is the sketch in the XY plane: http://www.wolframalpha.com/input/?i=plot+x^2+++y^2+-+x+=+0,+x^2+++y^2+=+1,+x+=+0,+y=+0

There are four lines of interest: two circles and two straight lines. The domain of the structure is defined by those lines. Can you see how?

 

[Saitama]

Here is the sketch in the XY plane: http://www.wolframalpha.com/input/?i=plot+x^2+++y^2+-+x+=+0,+x^2+++y^2+=+1,+x+=+0,+y=+0

There are four lines of interest: two circles and two straight lines. The domain of the structure is defined by those lines. Can you see how?

That's the same figure I drew myself. Can you please explain the meaning of "domain of structure", I think I am messong up with that word.

 

[voko]

The structure has a flat base at z = 0, i.e., it is in the XY plane. The contour of the base is somehow given by the four lines mentioned above. By integrating the structure's height within the domain described by the contour, you get the volume of the structure.

 

[Saitama]

The structure has a flat base at z = 0, i.e., it is in the XY plane. The contour of the base is somehow given by the four lines mentioned above. By integrating the structure's height within the domain described by the contour, you get the volume of the structure.

What's the structure's height?

Tell me if I am understanding the problem correctly. I need to find the volume of cylinder till the point it intercepts the paraboloid, right?

 

[jedishrfu]

What's the structure's height?

Tell me if I am understanding the problem correctly. I need to find the volume of cylinder till the point it intercepts the paraboloid, right?

I think that's right, you need to define an integral that is evaluated within the bounds of the circle base of the cylinder with the z ranging from 0 to whatever the parabola eqn says. The x would range from 0 to diameter of the cylinder and the y would range from 0 to whatever x^2 +y^2 -x = 0 defines it as and that by symmetry would give you have the volume.

 

[voko]

No, you are not understanding the problem correctly. The five surfaces must intersect to form some CLOSED volume. In the z direction it is simple: the bottom is the z = 0 plane, and the top is the paraboloid. What about the "sides" of the structure?

 

[Saitama]

I think that's right, you need to define an integral that is evaluated within the bounds of the circle base of the cylinder with the z ranging from 0 to whatever the parabola eqn says. The x would range from 0 to diameter of the cylinder and the y would range from 0 to whatever x^2 +y^2 -x = 0 defines it as and that by symmetry would give you have the volume.

No, you are not understanding the problem correctly. The five surfaces must intersect to form some CLOSED volume. In the z direction it is simple: the bottom is the z = 0 plane, and the top is the paraboloid. What about the "sides" of the structure?

Please see the attachment. Do I have to find the volume of the region filled with blue lines?

 

[jedishrfu]

No, you are not understanding the problem correctly. The five surfaces must intersect to form some CLOSED volume. In the z direction it is simple: the bottom is the z = 0 plane, and the top is the paraboloid. What about the "sides" of the structure?

Isn't the sides just the cylinder along the z-direction which is what the OP implied topped by the paraboloid surface?

 

[voko]

Please see the attachment. Do I have to find the volume of the region filled with blue lines?

Yes, that is correct. You still need to find the XY domain, however.

 

[voko]

Isn't the sides just the cylinder along the z-direction which is what the OP implied topped by the paraboloid surface?

Then the problem specifies too many unnecessary conditions such as x = 0 and y = 0. Look at the drawing in #8. You are suggesting that the inner circle is the base of the structure. Then surely x = 0 and y = 0 are completely superfluous as conditions. Yet they are specified.

 

[Saitama]

Yes, that is correct. You still need to find the XY domain, however.

Can you give me some hints for that? I am having trouble visualising the structure. The XY coordinates of the highest point of the base if (1/2,1/2).

 

[voko]

Look at the XY plane (as in #8). The left "side" is delimited by y = 0. What other sides should you select to form a closed base?

 

[Saitama]

Look at the XY plane (as in #8). The left "side" is delimited by y = 0. What other sides should you select to form a closed base?

There is no diagram in #8.
Do you mean #11? Isn't the left side delimited by x=0?

 

[voko]

Yes, I meant #11 and I meant x = 0 :)

 

[Saitama]

Yes, I meant #11 and I meant x = 0 :)

Are the other sides x=1, y=1/2 and y=-1/2?

 

[voko]

The other sides of the base must be intersections of the plane z = 0 with the other surfaces given. Those intersections are all given in #11.