# Volume of the following

• -EquinoX-
In summary, the task is to find the volume of a solid bounded by a paraboloid and a plane. One approach is to use Mathematica to plot the surfaces and make them transparent for better visualization. Then, use the polygon command to draw the plane and combine the plots using the show function. However, if Mathematica is not available, the volume can be found by dividing it into circular slices of thickness dx and integrating over dx. This is because the solid is a paraboloid, indicating that the volume can be calculated using the area of a circle.
-EquinoX-

## Homework Statement

Let W be the solid bounded by the paraboloid x = y^2 + z^2 and the plane x = 9.

How can I find the volume of this?

## The Attempt at a Solution

I find it hard to visualize so it makes me harder to find the volume... can someone help me?

-EquinoX- said:

## Homework Statement

Let W be the solid bounded by the paraboloid x = y^2 + z^2 and the plane x = 9.

How can I find the volume of this?

## The Attempt at a Solution

I find it hard to visualize so it makes me harder to find the volume... can someone help me?

Hello Equinox. The key to doing these effectively is to become proficient at plotting them. I think in Mathematica you can use:

ContourPlot3D[x==y^2+z^2,{x,-5,5},{y,-5,5},{z,-5,5}]

Ok, assume that gets the part you want. Next is to make it transparent so that you can better see the surfaces. Use "PlotStyle->{Opacity[0.5],LightPurple} or something like this.

Next is to draw the plane x=9. Do that with Polygon command and again use an opacity factor. Then combine the plots with Show[{p1,p2}]. Alright, the learning curve is slow at first, but once you get the hang of it, you can create a very nice visualization relatively quickly of this and much more complicated volume integrations and then the integrations become easy to set up once you have a nice picture. It's worth the effort. :).

Seems pretty complex.. I don't have mathematica as well

Hi -EquinoX-!

(try using the X2 tag just above the Reply box )
-EquinoX- said:
Let W be the solid bounded by the paraboloid x = y^2 + z^2 and the plane x = 9.

I find it hard to visualize so it makes me harder to find the volume... can someone help me?

It's a paraboloid …

you can tell because x= r2 is a parabola, so x = y2 + z2 is a parabola rotated about its principal axis.

Anyway, just divide the volume into circular slices of thickness dx, find the volume of each slice, and integrate.

I am pretty confused... should I integrate in polar, cylindrical, or spherical coordinate?

-EquinoX- said:
I am pretty confused... should I integrate in polar, cylindrical, or spherical coordinate?

Hi -EquinoX-!

I think you're making this over-complicated …

this isn't a ∫∫∫, where you have to decide whether it's dx dy dz or dr dθ dφ or dr dθ dz …

it's only a single ∫ because you know what the area of a circle is!

so, as I said, just use slices of thickness dx, and integrate (over dx)

Last edited:

## 1. What is volume and how is it measured?

Volume is the amount of space that an object takes up. It is measured in units such as cubic meters, liters, or cubic inches. The volume of an object can be determined by multiplying its length, width, and height.

## 2. How do you calculate the volume of a cube?

The volume of a cube can be calculated by using the formula V = s^3, where s represents the length of one of its sides. Simply cube the length of one side to find the volume.

## 3. Can the volume of irregularly shaped objects be measured?

Yes, the volume of irregularly shaped objects can be measured using the water displacement method. This involves placing the object in a container of water and measuring the change in water level to determine the volume of the object.

## 4. Is there a difference between volume and capacity?

Yes, volume and capacity are often used interchangeably, but there is a subtle difference between the two. Volume refers to the amount of space an object occupies, while capacity refers to the maximum amount of substance that can be held by an object.

## 5. What is the unit of measurement for volume in the metric system?

The unit of measurement for volume in the metric system is the cubic meter (m^3). However, smaller units such as milliliters (mL) and cubic centimeters (cm^3) are also commonly used.

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