Find volume between parabloid and parabolic sylinder

• MeMoses
In summary: The volume of the region is ##v = (y**2+z**2)^3##.Yes, those are correct. The volume of the region is ##v = (y**2+z**2)^3##.
MeMoses
EDIT - well its too late to change the title now, but I can spell corectly... sometimes.

Homework Statement

Find the volume of the region bounded by x = y**2 and z**2 and x = 2 - y**2

Homework Equations

Triple integrals and cylindricals

The Attempt at a Solution

I started by roughly graphing the two figures and I got stuck here for some reason. I tried cylindricals, but I figured there would be a better way. For whatever reason I can't get my head around the limits of integration for this problem. Solving the integral once I have it shouldn't be a problem, but I just need to setup the integral that's why my work here is limited. Thanks for any help

Hi MeMoses!
MeMoses said:
EDIT - well its too late to change the title now, but I can spell corectly... sometimes.

I think you meant a diabolic sylinder!
I tried cylindricals, but I figured there would be a better way. For whatever reason I can't get my head around the limits of integration for this problem. Solving the integral once I have it shouldn't be a problem, but I just need to setup the integral that's why my work here is limited.

One surface is of revolution, but the other isn't, so there's no short-cuts here.

You'll have to take slices of width dx perpendicular to the x-axis, and then slice those slices either "horizontally" (with width dz) or "vertically" (with width dy), and double-integrate

tiny-tim said:
Hi MeMoses! I think you meant a diabolic sylinder! One surface is of revolution, but the other isn't, so there's no short-cuts here.

You'll have to take slices of width dx perpendicular to the x-axis, and then slice those slices either "horizontally" (with width dz) or "vertically" (with width dy), and double-integrate
The real question is...
What's a Sylinder?

MeMoses said:
EDIT - well its too late to change the title now, but I can spell corectly... sometimes.

Homework Statement

Find the volume of the region bounded by x = y**2 and z**2 and x = 2 - y**2

What does the red highlighted definition mean? There isn't an equation after the word "and".

LCKurtz said:
What does the red highlighted definition mean? There isn't an equation after the word "and".

Hi LCKurtz!

"and" means "+"

tiny-tim said:
Hi LCKurtz!

"and" means "+"

How do you know he doesn't mean ##x = y^2## and ##x = z^2##?

the title?

tiny-tim said:
Hi LCKurtz!

"and" means "+"

LCKurtz said:
How do you know he doesn't mean ##x = y^2## and ##x = z^2##?

tiny-tim said:
the title?

Well, given that he obviously didn't post the problem word for word, I wouldn't make the assumption that the title is correct. Perhaps the problem is actually stated as I propose and the OP thinks that makes a paraboloid. No way of knowing until we see the correct wording one way or the other.

My bad, its x = y**2 + z**2 as thought

Would the limits
y**2+z**2 < x < 2-y**2,
-sqrt(1-(z**2)/2) < y <sqrt(1-(z**2)/2),
-sqrt(2) < z < sqrt(2)
get the correct volume?

MeMoses said:
Would the limits
y**2+z**2 < x < 2-y**2,
-sqrt(1-(z**2)/2) < y <sqrt(1-(z**2)/2),
-sqrt(2) < z < sqrt(2)
get the correct volume?

Yes, those are correct.

1.

What is the difference between a parabloid and a parabolic cylinder?

A parabloid is a three-dimensional shape that resembles a curved bowl, while a parabolic cylinder is a three-dimensional shape that resembles a curved tube. Both shapes have a parabolic cross-section, but the parabloid is curved in two directions while the parabolic cylinder is curved in one direction.

2.

How do you find the volume between a parabloid and a parabolic cylinder?

To find the volume between these two shapes, you would need to use calculus and the formula for the volume of a solid of revolution. This involves integrating the area of the cross-section of the two shapes as the variable of integration goes from 0 to the desired height of the solid.

3.

What real-life applications use the concept of finding volume between a parabloid and a parabolic cylinder?

The concept of finding volume between these two shapes is often used in architecture and engineering for designing structures with curved surfaces, such as domes and arches. It can also be applied in physics and fluid mechanics for calculating the volume of fluids in containers with curved bottoms.

4.

What are some challenges when calculating the volume between a parabloid and a parabolic cylinder?

One challenge is accurately determining the limits of integration and choosing the correct formula for the volume of a solid of revolution. Another challenge is ensuring the accuracy of the measurements used to calculate the volume, as small errors in measurement can lead to significant discrepancies in the final result.

5.

Are there any alternative methods for finding the volume between a parabloid and a parabolic cylinder?

Yes, there are alternative methods such as using geometric formulas to approximate the volume or using computer software to create a 3D model and calculate the volume. However, these methods may not be as accurate as using calculus and the formula for the volume of a solid of revolution.

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