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My main issue (multivariate calculus), I want to plot the body made within the surfaces

y + z = 4, y = 4 - x^2, y = 0 and z = 0.

the shape of which I am going to triple integrate as soon as I figure out how (not this topic but help is always welcome).

So far I have been trying using Plot3d, Contourplot3D and RegionFunction to show the area,

First try: 

Show[
  {Plot3D[{z = 4 - y, y = 4 - x^2}, {x, -2, 2}, {y, -1, 5}, 
     Mesh -> None], 
   ContourPlot3D[{z == 0, y == 0}, {x, -2, 2}, {y, -1, 5}, {z, -1, 7}]}, 
  PlotRange -> All, AxesLabel -> {x, y}, Mesh -> None]

Second try:

Show[
  Plot3D[{4 - y, 4 - x^2, 0,}, {x, -2, 2}, {y, 0, 4.1},
    PlotStyle -> 
      {{Blue, Opacity[0.7]}, {Yellow, Opacity[0.4]}, 
       {Green, Opacity[0.4]}, {Red, Opacity[0.4]}}, 
      AxesLabel -> Automatic, 
      Mesh -> None]]

surfaces

(Was going to use y = 0 in red but I can't seem to get that one going so I limited to y > 0)

Now adding a RegionFunction here could maybe possibly show the shape I am working on, but I would need to use some kind of conditions since the "roof" is not simply made of one function but a mix of two. Any ideas on how I could solve this issue?

the issue being that If I add say 

RegionFunction -> Function[{x, y, z}, 4 - x^2 > (4 - y)]

half of the shape will disappear.

computation

Does anyone have a good solution? As you can see from my code I am not proficient in Mathematica yet.

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  • $\begingroup$ Have you looked at RegionPlot3D[]? $\endgroup$ – J. M.'s technical difficulties♦ May 7 '16 at 15:07
  • $\begingroup$ Looking into it now =), lets see if I can figure it out, thanks $\endgroup$ – Celebrin May 7 '16 at 15:12
  • $\begingroup$ Is the desired result something like the output of this: RegionPlot3D[ 0 <= z <= 4 - y && 0 <= y <= 4 - x^2, {x, -2, 2}, {y, -1, 5}, {z, 0, 6}, Mesh -> None, PlotPoints -> 100, PlotStyle -> Directive[Yellow, Opacity[0.5]]]? $\endgroup$ – kglr May 7 '16 at 15:14
  • $\begingroup$ Yes definetely, I would wish on some differences in color to visualize better, but yes, this is more or less what I want, thanks alot!, now I just need to figure this out so I can replicate it. I still find it kinda hard to visualize in my head, not to mention which limits to put on x y z, , thanks for this =) $\endgroup$ – Celebrin May 7 '16 at 15:35
  • $\begingroup$ @Celebrin, with Plot3D you can use RegionFunction -> Function[{x, y, z}, 0 <= z <= Min[4 - x^2, (4 - y)]] to get the same region. $\endgroup$ – kglr May 7 '16 at 15:36
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Plot3D[{4 - y, 4 - x^2, 0}, {x, -2, 2}, {y, 0, 4.1}, 
 PlotStyle -> {{Blue, Opacity[0.7]}, {Yellow, Opacity[0.4]}, {Green,  Opacity[0.4]}}, 
 AxesLabel -> Automatic, Mesh -> None, 
 RegionFunction -> Function[{x, y, z}, 0 <= z <= Min[4 - x^2, (4 - y)]]]

Mathematica graphics

ParametricPlot3D:

ParametricPlot3D[{{x, y, ConditionalExpression[4 - x^2, 4 - x^2 <= (4 - y)]}, 
 {x, y, ConditionalExpression[4 - y, 4 - x^2 > (4 - y)]}, {x, y, 0},
 {x, 0, Min[y, 1] (4 - x^2)}}, 
 {x, -2, 2}, {y, 0, 4.1}, Mesh -> None,  PlotPoints -> 100,
 PlotStyle -> {Opacity[0.5, Blue], Opacity[0.5, Green], 
               Opacity[0.5, Yellow], Opacity[0.9, Red]}]

Mathematica graphics

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  • $\begingroup$ Great! =) that is really great =) , in my opinion a better visualisation, no? I am however getting syntax errors with that code and cannot seem to find the fault, are you running it in a Show[] or something? Expression "Plot3D[{4-y,4-x^2,0},{x,-2,2},{y,0,4.1},PlotStyle-><<1>>" has no closing "]". $\endgroup$ – Celebrin May 7 '16 at 16:02
  • $\begingroup$ @Celebrin, I agree; it is faster than RegionPlot3D too. I added the missing { that was causing the error error message. $\endgroup$ – kglr May 7 '16 at 16:23
  • $\begingroup$ Btw, any ideas if I wanted to bring in the y=0 plane too?, could I bring it into plot3d? $\endgroup$ – Celebrin May 7 '16 at 16:30
  • $\begingroup$ @Celebrin, i updated with an alternative approach using ParametricPlot3D that includes a way to depict the y==0 plane. $\endgroup$ – kglr May 7 '16 at 16:56
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r = ImplicitRegion[{y + z <= 4, y <= 4 - x^2, y >= 0 , z >= 0}, {x, y,
     z}];
i = HoldForm[Integrate[1, {x, -2, 2}, {y, 0, 4 - x^2}, {z, 0, 4 - y}]];
cp = ContourPlot3D[{z == 0, y == 0, 4 - y - x^2 == 0, 
    y + z == 4}, {x, -2, 2}, {y, 0, 4}, {z, 0, 4}, Mesh -> None, 
   ContourStyle -> {Red, Green, Blue, Orange}, 
   PlotLegends -> "Expressions"];
rp = RegionPlot3D[r, PlotPoints -> 100, Background -> Black];
TraditionalForm[Column[
  {r,
   Row[{i, "=", ReleaseHold[i]}],
   Row[{"Volume[r]= ", Volume[r]}],
   Row[{cp, rp}]}, Alignment -> Center]]

enter image description here

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