Volume of the following

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. :).

 

Hi -EquinoX-!

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

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.

 

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)