How to Calculate the Volume of a Solid Bounded by a Paraboloid and a Plane?

[-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?

Homework Equations

The Attempt at a Solution

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

 

[squidsoft]

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?

Homework Equations

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. :).

 

[-EquinoX-]

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

 

[tiny-tim]

Hi -EquinoX-!

(try using the X² tag just above the Reply box )

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.

 

[-EquinoX-]

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

 

[tiny-tim]

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)